1. Introduction
Denote by
the rooted tree of all finite 3-groups G with elementary bicyclic commutator quotient
, and let
be the infinite pruned subtree of
, where all descendants of capable non-metabelian vertices are eliminated. The main intention of this paper is to prove that the information content of the tree
can be reduced to a finite set of representatives with the aid of two kinds of periodicity.
• Firstly, the well-known virtual periodicity isomorphisms
between the finite depth-pruned branches
,
, of a coclass subtree
are refined to strict periodicity isomorphisms between complete branches which reduce the information content of the infinite coclass subtree to the finite union of pre-period
and first primitive period
. The virtual periodicity was proved by du Sautoy [1] and independently by Eick and Leedham-Green [2] for groups of any prime power order. The strict periodicity for
and type
is proved in the present paper.
• Secondly, evidence is provided of co-periodicity isomorphisms
between the infinite coclass forests
,
, which reduce the information content of the pruned tree
to the union of pre-period
and first primitive period
, consisting of the leading six coclass forests only. The discovery of this co-periodicity is the progressive innovation in the present paper.
Together with the coclass theorems of Leedham-Green [3] and Shalev [4] , which imply that each coclass forest
consists of a finite sporadic part
and a finite number of coclass trees
,
, each having a finite information content due to the strict periodicity, this shows that the pruned infinite subtree
of the tree
is described by finitely many representatives only.
We begin with a general theory of descendant trees of finite p-groups with arbitrary prime p in §2 and we explain the conceptual foundations of the virtual periodicity isomorphisms between the finite branches of coclass subtrees [1] [2] [5] and the recently discovered co-periodicity isomorphisms between infinite coclass forests in §3. The behavior of algebraic invariants of the tree vertices and their automorphism groups is described with simple transformation laws in §4. The graph theoretic preliminaries are supplemented by connections between depth, width, information content and numbers of immediate descendants in §5, identifiers of groups in §6, and precise definitions of mainlines and sporadic parts in §7. The main theorems are presented in §8.
Then we focus on the tree
of finite 3-groups G with abelianization
. The flow of our investigations is guided by §10 concerning the remarkable infinite main trunk
of certain metabelian vertices in
which gives rise to the top vertices of all coclass forests
,
, by periodic bifurcations and constitutes the germ of the newly discovered co-periodicity
of length two. To start with a beautiful highlight, we immediately celebrate the simple structure of the first primitive period
in §§11 and 12 and defer the somewhat arduous task of describing the exceptional pre-period
to the concluding §§13 and 14.
Finally, we point out that our theory, together with the investigations of Eick [6] , provides an independent verification and confirmation of all results about the metabelian skeleton
of the tree
in the dissertation of Nebelung [7] , since
is a subtree of the pruned tree
. The present paper shows the co-periodicity of the sporadic parts
and coclass trees
,
, of the coclass forests
, and ( [6] , §5.2, pp. 114-116) establishes the connection between the coclass trees
and infinite metabelian pro-3 groups of coclass r.
2. Descendant Trees and Coclass Forests
Let p be a prime number. In the mathematical theory of finite groups of order a power of p, so-called p-groups, the introduction of the parent-child relation by Leedham-Green and Newman ( [8] , pp. 194-195) has simplified the classification of such groups considerably. The relation is defined in terms of the lower central series
of a p-group G, where
(2.1)
in particular,
is the commutator subgroup of G. Since the series becomes stationary,
(2.2)
a non-trivial p-group
is nilpotent of class
.
Definition 2.1. If G is non-abelian, then the class-
quotient
(2.3)
is called the parent of G, and G is a child (or immediate descendant) of
.
Parent and child share a common class-1 quotient (or derived quotient or abelianization), since
(2.4)
according to the isomorphism theorem. The lower central series of
is shorter by one term:
(2.5)
and thus
.
Definition 2.2. For an assigned finite p-group
, the descendant tree
with root R is defined as the digraph
whose set of vertices V consists of all isomorphism classes of p-groups G with
, for some
, and whose set of directed edges E consists of all child-parent pairs
(2.6)
The mapping
is called the parent operator.
If the root R is abelian, then all vertices of the tree
share the common abelianization
. Since a nilpotent group with cyclic abelianization is abelian, the descendant tree
of a cyclic root
consists of the single isolated vertex R. The classification of p-groups by their abelianization is refined further, if directed edges are restricted to starting vertices G with cyclic last non-trivial lower central
of order p. Then the descendant tree of R splits into a countably infinite disjoint union
(2.7)
of directed subgraphs, where
and the vertices of the component
with fixed
share the same coclass,
, as a common invariant, since the logarithmic order
and the nilpotency class
of the parent
and child G satisfy the rule
(2.8)
Definition 2.3. Thus, the components
with
are called the coclass subgraphs of the descendant tree
.
According to the coclass theorems by Leedham-Green [3] and Shalev [4] , a coclass graph
is the disjoint union of a finite sporadic part
and finitely many coclass trees
(with infinite mainlines), that is, a forest for which there exist integers
such that
(2.9)
Definition 2.4. In the present paper, the focus will lie on finite p-groups with fixed prime
arising as descendants of the fixed elementary bicyclic 3-group
of order
with
, where
denotes the cyclic group of order n. This assumption permits a simplified notation by omitting the explicit mention of p and R. Further, we shall slightly reduce the complexity of the forests
,
, by eliminating the descendants of capable (i.e., non-terminal) non-metabelian vertices. This pruned light-weight version of
will be denoted by
, called the coclass-r forest, and Formula (2.9) becomes
(2.10)
and possibly different integers
and
.
Remark 2.1. In §§11 and 12 it will turn out that the coclass trees
with metabelian mainlines do not contain any capable non-metabelian vertices. So the pruning process from
to
concerns the sporadic part
, and reduces the number
of coclass trees by eliminating those with non-metabelian mainlines entirely, but does not affect the coclass trees with metabelian mainlines, which remain complete in spite of pruning.
3. Isomorphic Digraphs and Trees
In general, we denote a graph
as a pair
with set of vertices V and set of edges E.
Definition 3.1. Let
and
be two digraphs with directed edges in
, respectively
. If there exists a bijection
such that
(3.1)
then
and
are called isomorphic digraphs, and
is an isomorphism of digraphs.
When
is a finite digraph with vertex cardinality
, we can identify V with the set
. Then the set of directed edges
is characterized uniquely by the characteristic function
of E in
, which is called the
adjacency matrix
of
. Its entries are defined, for all
, by
(3.2)
Proposition 3.1. Let
and
be two finite digraphs with n vertices. Then
and
are isomorphic if and only if there exists a bijection
such that the entries of the adjacency matrices
coincide for all
.
Proof. The bijection
satisfies the condition in Formula (3.1) if and only if
⇔
⇔
⇔
.
The in-resp. out-degree of a vertex
in a finite digraph can be expressed in terms of the vth column-resp. row-sum of the adjacency matrix:
(3.3)
In particular, if
is a finite directed in-tree with root R, then each row of the adjacency matrix A corresponding to a vertex
contains a unique 1 and
(3.4)
Proposition 3.2. Let
and
be two rooted directed in-trees, and denote by
and
their parent operators. Then a bijection
with
is an isomorphism of rooted directed in-trees if and only if
for all
, that is,
(briefly:
commutes with the parent operator), as shown in Figure 1.
Proof. Recall that each row of the adjacency matrix A of the tree
corresponding to a vertex
,
, contains a unique 1. This fact can be used to define the parent operator
of
by
⇔
. Consequently, if
has the claimed property to commute with the parent operator, then
⇔
⇔
⇔
⇔
⇔
⇔
. For infinite trees, the steps concerning adjacency matrices must be omitted. The proof of the converse statement is similar.
4. Algebraically Structured Digraphs
4.1. General Invariants and Their Transformation Laws
Since the vertices of all trees and branches in this paper are realized by isomorphism classes of finite p-groups, the abstract intrinsic graph theoretic structure of the trees and branches can be extended by additional concrete structures defined with the aid of algebraic invariants of p-groups.
Not all algebraic structures are strict invariants under graph isomorphisms. Some of them change in a well defined way, described by a mapping
, the transformation law, when a graph isomorphism is applied. This behaviour is made precise in the following definitions.
Definition 4.1. Let
be a graph. Suppose that
is a set, and each vertex
is associated with some kind of information
. Then
is called a structured graph with respect to the mapping
,
.
If
and
are two structured digraphs with respect to mappings
,
, and
,
, and
is a mapping, then an isomorphism of digraphs
is called a
-iso- morphism of structured digraphs
and
, if
for all
, that is,
, as visualized in Figure 2.
In particular, if the sets
coincide and
is the identity mapping of the set X, then
is called a strict isomorphism of structured digraphs, and it satisfies the relation
.
Definition 4.2. Let
and
be two structured digraphs with structure mappings
and
, and let
be a
-isomorphism of the two structured digraphs with respect to a mapping
, that is,
. Then
is called a
-invariant under
(or invariant under the isomorphism
and transformation law
). In particular, if
,
and
, then
is called a strict invariant under
.
4.2. Algebraic Invariants Considered in This Paper
With respect to applications in other mathematical theories, in particular, algebraic number theory and class field theory, certain properties of the
automorphism group
of a finite 3-group G are crucial. The general frame of these aspects is the following.
Definition 4.3. Let p be an odd prime number and let G be a pro-p group. We call G a group with GI-action or a
-group, if there exists a generator inverting automorphism
such that
, for all
, or equivalently
, for all
. If additionally
, for all
, then G is called a group with RI-action or group with relator inverting automorphism. If
contains a bicyclic subgroup
, then we call G a group with
-action. It is convenient to define the action flag of G by
(4.1)
Remark 4.1. Suppose that G is a finite p-group with odd prime p. We point out that 2 divides the order
, if G is a group with GI-action, but the converse claim may be false. If G is a group with
-action, then 4 divides
, but we emphasize that the converse statement, even in the case that 8 divides
, may be false, when
contains a cyclic group
or a (generalized) quaternion group
of order
with
.
For a brief description of abelian quotient invariants in logarithmic form, we need the concept of nearly homocyclic p-groups. With an arbitrary prime
these groups appear in ( [9] , p. 68, Thm. 3.4) and they are treated systematically in ( [7] , 2.4). For our purpose, it suffices to consider the special case
.
Definition 4.4. By the nearly homocyclic abelian 3-group
of order
, for an integer
, we understand the abelian group with logarithmic type invariants
, where
with integers
and
, by Euclidean division with remainder. Additionally, including two degenerate cases, we define that
denotes the cyclic group
of order 3, and
denotes the trivial group 1.
The following invariants
of finite 3-groups
with abeliani- zation
will be of particular interest in the whole paper:
• The logarithmic order
,
,
• The nilpotency class
, connected with the index of nilpotency m by the relation
, where the lower central series stops with
,
• The coclass
, defined by
,
• The order of the automorphism group
,
,
• The action flag
, defined by Formula (4.1),
• The transfer kernel type (TKT)
,
, where
denote the transfer homomorphisms from v to the maximal subgroups
, for
,
• The transfer target type (TTT)
,
, viewed as abelian quotient invariants, where
denote the maximal subgroups of v ( [10] , Dfn. 5.3, p. 83),
• The abelian quotient invariants of the first TTT component
,
, where k denotes the defect of commutativity of v ( [11] , 2, p. 469),
• The abelian quotient invariants of the commutator subgroup
,
(or
in irregular cases) ( [7] , Satz 4.2.4, p. 131),
• The relation rank
,
, which coincides with the rank of the p-multiplicator of v ( [13] , Thm. 2.4),
• The nuclear rank
, i.e. the rank of the nucleus of
( [13] , Thm. 2.4). For a coclass tree, the nuclear rank is given by
.
Remark 4.2. Abelian quotient invariants are given in logarithmic notation. The transfer kernel type
is simplified by a family of non-negative integers, in the following way: for
,
(4.2)
5. The Graph Theoretic Structure of a Tree
Cardinality of Branches and Layers, Depth and Width of a Tree
The graph theoretic structure of a coclass tree
with unique infinite mainline and finite branches, consisting of isomorphism classes of finite p-groups, is described by the following concepts.
Definition 5.1. Let
be a coclass tree. Suppose that the tree root R is of logarithmic order
, and denote by
the unique mainline vertex with
. In particular,
.
For
, the difference set
is called the eth branch of
.
Let
be one of the branches of
. For any integer
, we let
(5.1)
denote the nth layer of
, respectively
.
The width of the tree is the maximal cardinality of its layers,
(5.2)
Each vertex v of the branch
is connected with the mainline by a unique finite path of directed edges from v to the branch root
, formed by the iterated parents
of v,
(5.3)
The length
of this path is called the depth
of v.
The depth of a branch
is the maximal depth of its vertices,
(5.4)
Definition 5.2. Let
be a coclass tree. The depth of the tree is the maximal depth of its branches,
(5.5)
Throughout this paper, we assume that both, the depth
and the width
of the tree, are bounded. This assumption is satisfied by all trees of finite 3-groups under investigation in the sequel. However, we point out that that tree
of finite 5-groups with coclass one has unbounded depth, and the tree
of finite 7-groups with coclass one even has unbounded width and depth. (Compare [6] , 5.1, pp. 113-114)
Lemma 5.1. Let
,
and
. Then
(5.6)
Proof. Since
is the root of the branch
, we have
, but
. Since
, there exists a vertex
, necessarily terminal if
, such that
. The iterated parents
of t form the unique finite path from t to the branch root
(see Figure 3),
![]()
and we have
but
.
Lemma 5.2. Let
and
. Then
(5.7)
Proof. Since
, we have
for each
. A branch
with
cannot contribute to
. On the other hand, if
, then a branch
with
cannot contribute to
either, since
, according to Lemma 5.1, and we obtain
(see Figure 3). Consequently,
![]()
Figure 3. Schematic coclass tree
with ultimately periodic branches and layers.
(5.8)
Since the implementation of the p-group generation algorithm [12] [13] [14] in the computational algebra system MAGMA [15] [16] [17] is able to give the number of all, respectively only the capable, immediate descendants (children) of an assigned finite p-group, we express the cardinalities of the branches of a coclass tree, which were given in a preliminary form in Lemma 5.1, in terms of these numbers
, respectively
.
Theorem 5.1. Let
be a coclass tree with tree root R of logarithmic order
, pre-period of length
, and period of primitive length
. For each vertex
, denote by
the number of all children (of step size
) and by
the number of capable children of v. When
is the vertex with
on the mainline of
, let
with
be the capable children of
, in particular, let
be the next mainline vertex. Finally, let
with
denote the capable children of
, for each
.
1) If the tree is of depth
, then
(5.9)
2) If the tree is of depth
, then
(5.10)
3) If the tree is of depth
, then
(5.11)
Proof. Put
. Generally, we have
with
, according to Lemma 5.1.
If
, then
and
. We have
and
, since the next mainline vertex
is one of the
children of
but does not belong to
. Thus, we obtain
.
If
, then
and
, where
and
as before, and
. Therefore,
.
If
, then
and
, where
,
,
as before, and
. Thus,
![]()
Remark 5.1. In Theorem 5.1, item (1) is included in item (2), since
implies
, and item (2) is included in item (3), since
implies
, for all
.
Corollary 5.1. Under the same assumptions as in Theorem 5.1, the width of the coclass tree
, in dependence on the depth
and the periodicity
, is generally given by
(5.12)
For assigned small values of the depth
, the width can be expressed in terms of descendant numbers in the following manner:
1) If the tree is of depth
, then
(5.13)
2) If the tree is of depth
, then
is the maximum among the number
and all expressions
(5.14)
where n runs from
to
.
3) If the tree is of depth
, then
is the maximum among the numbers
,
, and all expressions
(5.15)
where n runs from
to
.
Proof. According to [1] [2] [5] , the periodicity of the branches of a coclass tree
with root
and bounded depth
and width
can be expressed by means of isomorphisms between branches, starting from the periodic root
:
(5.16)
where
denotes the length of the pre-period and
is the primitive period length. With Lemma 5.1, an immediate consequence is the periodicity of branch layer cardinalities:
![]()
According to Lemma 5.2, we have
, and thus
![]()
For finding the maximal layer cardinality, the root term
can be omitted, since each layer contains a mainline vertex. Beginning with
, the expression for the tree layer cardinality
is a sum of
terms and we must find the logarithmic order
where periodicity of all terms sets in. This leads to the inequality
with solution
. Consequently,
is the biggest logarithmic order for which a new value of the tree layer cardinality
may occur (see Figure 3). At the logarithmic order
, periodic repetitions of the values of tree layer cardinalities begin.
In the special case of
, Theorem 5.1 yields an expression in terms of descendant numbers:
![]()
The following concept provides a quantitative measure for the finite infor- mation content of an infinite tree with periodic branches.
Definition 5.3. By the information content of a coclass tree
we understand the sum of the cardinalities of all branches belonging to the pre-period and to the primitive period of
,
(5.17)
where
denotes the logarithmic order of the periodic root P of
(see Figure 3).
6. Identifiers of the SmallGroups Library
Independently of being metabelian or non-metabelian, a finite 3-group G of order up to
will be characterized by its absolute identifier
, according to the SmallGroups Database [18] [19] . Starting with order
, a group G is characterized by the absolute identifier of the parent
in the SmallGroups Database [19] together with a relative identifier
generated by the ANUPQ package [20] of MAGMA [17] . Here, s denotes the step size of the directed edge
. Occasionally, certain groups of order
and coclass 2 are identified by single capital letters
similarly as in [21] [22] [23] .
7. Mainlines of Coclass Trees and Sporadic Parts of Coclass Forests
If we define a mainline as a maximal path of infinitely many directed edges of step size
, then there arises the ambiguity that a vertex could be root of several coclass trees. The metabelian 3-group
, for instance, would be the end vertex of more then one mainline, namely on the one hand of the metabelian mainline
![]()
and on the other hand of non-metabelian mainlines, one which ends with
![]()
and three which end with
,
.
Therefore, an additional condition is required in the precise definition of a mainline.
Definition 7.1. A mainline is a maximal path of infinitely many equally oriented edges of step size
, in none of whose vertices other infinite paths of step size
are ending.
The end vertex of a mainline is called the root of a coclass tree.
Definition 7.1 can be expressed equivalently in terms of infinite pro-p groups ( [6] , 3.1, p. 107).
Example 7.1. The metabelian 3-group
is root of the coclass-2 tree
with metabelian mainline.
The metabelian 3-group
is root of the coclass-2 tree
with non-metabelian mainline. According to our pruning convention that descendants of capable non-metabelian vertices do not belong to the coclass forests
, this tree is not an object of examination in the present paper.
Finally, the metabelian 3-group
is not root of a coclass tree.
Based on the precise definition of a mainline and a root of a coclass tree, we are now in the position to give an exact specification of the sporadic part of a coclass forest.
Definition 7.2. The sporadic part of the coclass forest
with
is the complement of the union of the (finitely many) coclass trees in the forest,
(7.1)
There is no necessity, to restrict the concepts of a mainline, a coclass tree and its root further by stipulating the coclass stability of the root. It is therefore admissible that directed edges of step size
end in vertices (mainline or of depth
) of a coclass tree, due to the phenomenon of multifurcation.
Example 7.2. In the second mainline vertex
of the coclass-2 tree
with root
, a bifurcation occurs, due to the nuclear rank
. In fact, the directed edge of step size
which ends in the vertex
is the final edge of an infinite path with alterating step sizes
and
, due to periodic bifurcations. However, this non-meta- belian path is not the topic of investigations in the present paper. For detailed information on these matters see [24] and [25] .
The same is true for the second mainline vertex
of the coclass-2 tree
with root
.
The unnecessary requirement of coclass stability would eliminate the pre-periods of the trees
and
and enforce purely periodic subtrees with periodic coclass-settled roots, namely
and
.
8. Two Main Theorems on Periodicity and Co-Periodicity Isomorphisms
An important technique in the theory of descendant trees is to reduce the structure of an infinite tree to a periodically repeating finite pattern. In particular, it is well known [1] [2] [5] that an infinite coclass tree
of finite p-groups with fixed coclass
is the disjoint union of its branches
, which can be partitioned into a single finite pre-period
of length
and infinitely many copies of a finite primitive period
of length
, where the integer
characterizes the position of the periodic root on the mainline, provided the tree is suitably depth-pruned.
The following first main result of this paper establishes the details of the primitive period of branches of five coclass-4 trees
, with
, respectively of three coclass-5 trees
, with
, of finite 3-groups with mainline vertices having a single total transfer kernel and roots
with
, respec- tively
with
, writ- ten in the notation of [18] [19] [20] . In fact, we prove more than the virtual periodicity for arbitrary finite p-groups in [1] [2] [5] , since all trees of the particular finite 3-groups in our investigation have bounded depth and therefore reveal strict periodicity.
Theorem 8.1. (Main Theorem on Strict Periodicity Isomorphisms of Branches.)
For each integer
, there exists a bijective mapping
which is a strict isomorphism of finite structured in-trees for the strict invariants in-degree
, out-degree
, coclass
, relation rank
, nuclear rank
, action flag
, and transfer kernel type
. Moreover,
is a f-isomorphism of finite structured in-trees for the following
-invariants with their transformation laws
:
• logarithmic order
with
,
• nilpotency class
with
,
• order of the automorphism group
with
,
• first component of the transfer target type
with
, and
• commutator subgroup
with
, respectively
.
Consequently, the branches of each tree
, respectively
, are purely periodic with primitive length at most
.
Proof. The
-isomorphisms between the finite branches of a tree describe the first periodicity and reduce an infinite tree to its finite primitive period, provided the periodicity is pure. This will be proved for even coclass
in Theorem 11.3 for
, in Thm. 11.4 for
, in Thm. 11.5 for
, in Thm. 11.6 for
, and in Thm. 11.7 for
. For odd coclass
, it will be proved in Theorem 12.3 for
, in Thm. 12.4 for
, and in Thm. 12.5 for
.
Invariants connected with the nilpotency class are not strict and satisfy the following transformation laws: the shift
for
and
, and the corresponding transformations
for
,and
for
, with fixed coclass r. For
, the transformation law is described by the homothety
.
Theorems 11.3, 11.4, 11.5, 11.6, 11.7 and 12.3, 12.4, 12.5 will give detailed descriptions of the structure of these trees, in particular they will establish a quantitative measure for the finite information content of each tree.
Remark 8.1. According to Theorem 8.1, the diagrams of coclass-r trees
whose mainline vertices V possess a single total kernel
among the transfers
to the four maximal subgroups
reveal several surprising features: firstly, the branches are purely periodic of primitive length at most 2 without pre-period, secondly, the branches are of uniform depth 2 only, and finally, none of the vertices gives rise to descendants of coclass bigger than r. So the trees are entirely regular and coclass-stable, in contrast to the trees with 3-groups G of coclass
as vertices.
Unfortunately it is much less well known that the entire metabelian skeleton
of the descendant tree
of the elementary bicyclic 3-group
is the disjoint union of its coclass subgraphs
, where each component
consists of a finite sporadic part
and finitely many metabelian coclass trees
, and there is a periodicity
for each
. This was proved by Nebelung [7] and confirmed by Eick ( [6] , Cnj. 14, p. 115).
The following second main result of this paper extends the periodicity from the metabelian skeleton to the entire descendant tree, including all the non-metabelian vertices, provided the mainline vertices are still metabelian. Here, we include coclass trees of finite 3-groups with mainline vertices having two total transfer kernels and roots
with
, respectively
with
.
Theorem 8.2. (Main Theorem on Co-Periodicity Isomorphisms of Coclass Trees.)
Let the integer
be an upper bound. For each integer
, and for each of the six roots
,
, with even coclass
, respectively the four roots
,
, with odd coclass
, there exists a bijective mapping
, respectively
, which is a strict isomorphism of infinite structured in-trees for the strict invariants in-degree
, out-degree
, relation rank
, nuclear rank
, action flag
, and transfer kernel type
. Moreover,
is a f-isomorphism of infinite structured in-trees for the following f-invariants with their transformation laws f:
• logarithmic order
with
,
• nilpotency class
with
,
• coclass
with
,
• order of the automorphism group
with
,
• first component of the transfer target type
with
, and
• commutator subgroup
with
, respectively
.
Proof. The statement for the metabelian skeletons
of the coclass trees
is one of the main results of Nebelung’s thesis [7] . With the aid of Theorem 8.1, the periodicity of the entire coclass trees
with
and fixed subscript i has been verified by computing the metabelian and non-metabelian vertices of the first four branches
with
of the trees
. The computations were executed by running our own program scripts for the Computer Algebra System MAGMA [17] , which contains an implementation of the p-group generation algorithm by Newman [12] [26] and O’Brien [13] [14] , the SmallGroups Database [18] [19] , and the ANUPQ package [20] . It turned out that, firstly,
and
, for each
, and secondly,
and
, for each
.
The established
-isomorphisms between the infinite coclass trees
and
, for
, describe the germ of the second periodicity expressed in Conjecture 8.1. Invariants connected with the nilpotency class or coclass are not strict and are subject to the following mappings: the shifts
for
,
for
,and
for
,and the corresponding transformations
for
,and
for
.For
,the transformation law is described by the homothety
.
Thus, the confidence in the validity of the following conjecture is supported extensively by sound numerical data.
Conjecture 8.1. (Co-Periodicity Isomorphisms of All Coclass-r Trees for
.)
Theorem 8.2 remains true when the upper bound
is replaced by any upper bound
.
Consequently, all coclass trees
with
and fixed subscript i are co-periodic in the variable coclass parameter r with primitive length
. The eight coclass trees
with
, and
for
,
for
,
for
, can be viewed as the pre-period of the co-periodicity. (Compare [6] , Cnj. 14, p. 115).
9. Parametrized Polycyclic Power-Commutator Presentations
The general graph theoretic and algebraic foundations of the coclass forests
with
have been developed completely in the preceding Sections 2 - 7. Now we can turn to the main goal of the present paper, that is, the proof of the main theorems in section 8 by the systematic investigation of finite 3-groups G with commutator quotient
, represented by vertices of the descendant tree
, with the single restriction that the parent
of G is metabelian. To this end, we first need parametrized presentations for all metabelian vertices of
.
9.1. 3-Groups of Coclass r = 1
The identification of 3-groups G with coclass
, which are metabelian without exceptions [27] , will be achieved with the aid of parametrized polycyclic power-commutator presentations, as given by Blackburn [9] :
(9.1)
where
and
are bounded parameters, and the index of nilpotency
is an unbounded parameter.
9.2. 3-Groups of Coclass r ≥ 2
Metabelian 3-groups with coclass
will be identified with the aid of parametrized polycyclic power-commutator presentations, given by Nebelung [7] :
(9.2)
where
are bounded parameters, and the index of nilpo- tency
,the logarithmic order
,and the CF-invariant
are unbounded parameters.
10. The Backbone of the Tree
: The Infinite Main Trunk
The flow of our investigations is guided by the present section concerning the remarkable infinite main trunk
of certain metabelian vertices in
which gives rise to the top vertices of all coclass forests
by periodic bifurcations and constitutes the germ of the newly discovered co-periodicity
of length two. Since the minimal possible values of the nilpotency class and logarithmic order of a finite metabelian 3-group with coclass
, belonging to the forest
, are given by
and
, it follows that G must be an immediate descendant of step size
of its parent
. The crucial fact is that this parent is precisely the vertex
with
of the main trunk. In the following, we rather use the coclass j of the parent than r of the children.
Theorem 10.1. (The main trunk.)
1) In the descendant tree
of the abelian root
, there exists a unique infinite path of (reverse) directed edges
such that, for each fixed coclass
, every metabelian 3-group G with
and
is a proper descendant of
.
2) The trailing vertex
is exactly the extra special Blackburn group
with exceptional transfer kernel type (TKT ) a.1,
.
3) All the other vertices
with
share the common TKT b.10,
, possess nilpotency class
, coclass
, logarithmic order
, abelian commutator subgroup of type
, and transfer target type
, where
.
4) For
, periodicity of length 2 sets in,
has nuclear rank
, relation rank
, and immediate descendant numbers (including non-metabelian groups)
![]()
Restricted to metabelian groups, the immediate descendant numbers are
![]()
All immediate descendants are
-groups, if
is odd, but only
, if
, and
, if
is even.
Proof. See the dissertation of Nebelung ( [7] , p. 192).
Remark 10.1. Although the number of metabelian children of step sizes
of the vertex
with
fit into the periodic pattern
, the number of all children of step sizes
of
is bigger than usual with
instead of
. Therefore, periodicity starts with
and not with
.
Corollary 10.1. (All coclass trees with metabelian mainlines.)
The coclass trees of 3-groups G with
, whose mainlines consist of metabelian vertices, possess the following remarkable periodicity of length 2, drawn impressively in Figure 4.
1) For even
, the vertex
with subscript
of the main trunk has exactly 4 immediate descendants of step size
giving rise to coclass trees
whose mainline vertices are metabelian 3-groups G with odd
and fixed TKT, either d.19,
, or d.23,
, or d.25,
, or b.10,
, the latter with root
.
2) For odd
, the vertex
with subscript
of the main trunk has exactly 6 immediate descendants of step size
giving rise to coclass trees
whose mainline vertices are metabelian 3-groups G with even
and fixed TKT, either d.19,
, twice, or d.23,
, or d.25,
, twice, or b.10,
, the latter with root
.
3) The unique pre-periodic exception is the vertex
of the main trunk, which has exactly 3 immediate descendants of step size
giving rise to coclass trees
whose mainline vertices are metabelian 3-groups G with even
and fixed TKT, either c.18,
, or c.21,
, or b.10,
, the latter with root
.
Proof. See the dissertation of Nebelung ( [7] , 5.2, pp. 181-195).
![]()
Figure 4. Metabelian mainline skeleton of the descendant tree
.
11. Sporadic and Periodic 3-Groups G of Even Coclass ![]()
Although formulated for the particular coclass
, all results for periodic groups and most of the results for sporadic groups in this section are valid for any even coclass
. The only exception is the bigger (and thus pre-periodic) sporadic part
of the coclass forest
, described in Proposition 11.2, whereas the (co-periodic) standard case, the sporadic part
of the coclass forest
, is presented in Proposition 11.1.
Figure 5 sketches an outline of the metabelian skeleton of the coclass forest
in its top region. The vertices
and
,
![]()
Figure 5. Metabelian interface between the coclass forests
and
.
with the crucial bifucation from
to
, belong to the infinite main trunk (§10).
Proposition 11.1 (Co-periodic standard case.)
The sporadic part
of the coclass-6 forest
consists of
• 13
isolated metabelian vertices of order 313 with types F.7, F.11, F.12, F.13,
• 8
metabelian roots of finite trees with types G.16, G.19, H.4, together with a metabelian child having a GI-action, which is unique for each root, and 22 metabelian and 38 non-metabelian children without GI-action, all with depth
and
,
• 66
isolated vertices with
and types d.19, d.23, d.25,
• 179 isolated vertices with
and type b.10,
• 23 capable vertices with
and type b.10,
Whose children do not belong to
, by definition.
The action flag of all metabelian top vertices with depth
is
. The value
only occurs for all vertices with type b.10, d.25, G.19, and certain vertices with type G.16, H.4, but never for type d.19, d.23, F.7, F.11, F.12, F.13. Exactly the isolated vertices with depth
have an RI-action.
Together with the 6 metabelian roots
,
, of coclass-6 trees, the
top vertices of depth
of
are exactly the
children of step size
of the main trunk vertex
, and the
capable vertices among them correspond to the invariant
of
.
Proposition 11.2. (Pre-periodic exception.)
The constitution of the sporadic part
of the coclass-4 forest
with respect to the 21 metabelian top vertices and their 68 children (here with order 39, resp.
) is the same as described for
in Proposition 11.1, but the number of non-metabelian top vertices of depth
is bigger, namely
• 88
isolated vertices with
and types d.19, d.23, d.25,
• 12
capable vertices with
and types d.19, d.23, d.25,
whose children do not belong to
, by definition,
• 268 isolated vertices with
and type b.10,
• 58 capable vertices with
and type b.10,
whose children do not belong to
, by definition.
The distribution of the action flags
is the same as in Proposition 11.1, but the total census of top vertices is considerably bigger:
Together with the 6 metabelian roots
,
, of coclass-4 trees, the
top vertices of depth
of
are exactly the
children of step size
of the main trunk vertex
, and the
capable vertices among them corres- pond to the invariant
of
.
Theorem 11.1. The coclass-r forest
with any even
is the disjoint union of its finite sporadic part
with total information content
(11.1)
and
infinite coclass-r trees
with roots
, where
![]()
The algebraic invariants for groups with positive action flag
, and in cumulative form for
, are given for
in Table 1, where the parent vertex
on the main trunk is also included, but the 426 non-metabelian top vertices of depth
are excluded.
Proof. (of Propositions 11.1, 11.2, and Theorem 11.1) We have computed the sporadic parts
of coclass forests
with even
up to
by means of MAGMA [17] . Except for the differences pointed out in the
![]()
Table 1. Data for sporadic 3-groups G with
in the forest
.
Propositions 11.1 and 11.2, they all share a common graph theoretic structure with
. The forest
contains 6 roots of infinite coclass trees with metabelian mainlines (a unique root
of type b and five roots
of type d), namely
,
,
,
,
,
,
which give rise to the periodic part of
, and 51 sporadic metabelian groups of type F, G or H. Among the groups of the sporadic part
, there are 13 isolated metabelian vertices with type F, and 8 metabelian roots of finite trees with type G or H and tree depth 1, each with a unique metabelian child having
. The other
children with
, of which
are metabelian and
have derived length 3, are omitted in the forest diagram, Figure 5. Additionally,
, respectively
, contains 426, respectively 268, non-metabelian top vertices, which gives a total information content
of
, respectively
, representatives. The difference is an excess of
vertices in
.
The metabelian skeleton of both,
and
, consists of
vertices. The results for metabelian groups are in accor- dance with the fourth tree diagram
,
, in ( [7] , fourth double page between pp. 191-192). The metabelian groups in Table 1 correpond to the representatives of isomorphism classes in ( [28] , pp. 36-38 and 42-45).
11.1. The Unique Mainline of Type b.10* for Even Coclass r ≥ 4
Proposition 11.3. (Periodicity and descendant numbers.)
The branches
,
, of the coclass-4 tree
with mainline vertices of transfer kernel type b.10*,
, are periodic with pre-period length
and with primitive period length
, that is,
are isomorphic as digraphs, for all
.
The graph theoretic structure of the tree is determined uniquely by the numbers
of immediate descendants and
of capable immediate descen- dants of the mainline vertices
with logarithmic order
:
for the root
with
,
for all mainline vertices
with even logarithmic order
,
for all mainline vertices
with odd logarithmic order
.
Proof. (of Proposition 11.3) The statements concerning the numbers
of immediate descendants of the mainline vertices
with
have been obtained by direct computation with MAGMA [17] , where the p-group generation algorithm by Newman and O'Brien [12] [13] [14] is implemented. In detail, we proved that there are:
4, resp. 6, metabelian vertices with bicyclic centre
, resp. cyclic centre
, and
5, resp. 6, non-metabelian vertices with
, resp.
,
together 21 vertices (10 of them metabelian) in the pre-periodic branch
,
and the primitive period
of length
consists of
6, resp. 6, metabelian vertices with
, resp.
, and
9, resp. 9, non-metabelian vertices with
, resp.
,
together 30 vertices (12 of them metabelian) in branch
, and
4, resp. 8, metabelian vertices with
, resp.
, and
5, resp. 10, non-metabelian vertices with
, resp.
,
together 27 vertices (12 of them metabelian) in branch
.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung ( [7] , Thm. 5.1.16, pp. 178-179, and the fourth Figure, e ≥ 5, e
1, on the double page between pp. 191-192). The tree
corresponds to the infinite metabelian pro-3 group
in ( [6] , Cnj. 15 (b), p. 116). Although every branch contains 12 metabelian vertices, the primitive period length is
rather than
, even for the metabelian skeleton, since the constitution
of branch
is different from
for branch
, as proved above.
The claim of the virtual periodicity of branches has been proved generally for any coclass tree by du Sautoy [1] , and independently by Eick and Leedham- Green [2] . Here, the strict periodicity was confirmed by computation up to branch
and undoubtedly sets in at
.
Theorem 11.2. (Graph theoretic and algebraic invariants.)
The coclass-4 tree
of finite 3-groups G with coclass
which arises from the metabelian root
has the fol- lowing graph theoretic properties.
1) The pre-period
of length
is irregular.
2) The cardinality of the irregular branch is
.
3) The branches
,
, are periodic with primitive period
of length
.
4) The cardinalities of the regular branches are
and
.
5) Depth, width, and information content of the tree are given by
(11.2)
The algebraic invariants of the groups represented by vertices forming the pre-period
and the primitive period
of the tree are given in Table 2. The leading six branches
are drawn in Figure 6.
Remark 11.1. The algebraic information in Table 2 is visualized in Figure 6. By periodic continuation, the figure shows more branches than the table but less details concerning the exact order
of the automorphism group.
Proof. (of Theorem 11.2) According to Proposition 11.3, the logarithmic order of the tree root, respectively of the periodic root, is
, respectively
.
Since
for all mainline vertices
with
, according to
![]()
Figure 6. The unique coclass-4 tree
with mainline of type b.10*.
Proposition 11.3, the unique capable child of
is
, and each branch has depth
, for
. Consequently, the tree is also of depth
.
With the aid of Formula (5.9) in Theorem 5.1, the claims (2) and (4) are consequences of Proposition 11.3:
,
, and
.
According to Formula (5.13) in Corollary 5.1, where n runs from
to
, the tree width is the maximum
of the expressions
,
, and
.
![]()
Table 2. Data for 3-groups G with
of the coclass tree
.
The information content of the tree is given by Formula (5.17) in the Definition 5.3:
![]()
The algebraic invariants in Table 2, that is, depth
, derived length
, abelian type invariants of the centre
, relation rank
, nuclear rank
, abelian quotient invariants
of the first maximal subgroup, respectively
of the commutator subgroup, transfer kernel type
, action flag
, and the factorized order
of the automorphism group have been computed by means of program scripts written for MAGMA [17] .
Each group is characterized by the parameters of the normalized representative
of its isomorphism class, according to Formula (9.2), and by its identifier
in the SmallGroups Database [19] .
The column with header # contains the number of groups with identical invariants (except the presentation), for each row.
Corollary 11.1. (Actions and relation ranks.) The algebraic invariants of the vertices of the structured coclass-4 tree
are listed in Table 2. In particular:
1) The groups with
-action are all mainline vertices
,
, the two terminal vertices
with odd
, the two terminal vertices
with even
, and two terminal non-metabelian vertices with odd
.
2) With respect to the kernel types, all mainline groups of type b.10*,
, the two leaves of type d.25,
, with every odd logarithmic order, two distinguished metabelian leaves of type b.10 with every even logarithmic order, and two distinguished non-metabelian leaves of type b.10 with every odd logarithmic order possess a
-action.
3) The relation rank is given by
for the mainline vertices
,
, and
otherwise. There do not occur any RI-actions.
Proof. (of Corollary 11.1) The existence of an RI-action on G has been checked by means of an algorithm involving the p-covering group of G, written for MAGMA [17] . The other claims follow immediately from Table 2, continued indefinitely with the aid of the periodicity in Prop. 11.3.
11.2. Two Mainlines of Type d.19* for Even Coclass r ≥ 4
Proposition 11.4. (Periodicity and descendant numbers.)
The branches
,
, of the first coclass-4 tree
with mainline vertices of transfer kernel type d.19*,
, are purely periodic with primitive length
and without pre-period,
, that is,
are isomorphic as digraphs, for all
.
The graph theoretic structure of the tree is determined uniquely by the numbers
of immediate descendants and
of capable immediate descendants of the mainline vertices
with logarithmic order
and of capable vertices v with depth 1 and
:
for mainline vertices
with odd logarithmic order
,
for mainline vertices
with even logarithmic order
,
for the capable vertex v of depth 1 and even logarithmic order
,
for two capable vertices v of depth 1 and odd logarithmic order
.
Proof. (of Proposition 11.4) The statements concerning the numbers
of immediate descendants of the mainline vertices
with
, and
of vertices with depth
and logarithmic order
, have been obtained by direct computation with the p-group generation algorithm [12] [13] [14] in MAGMA [17] . In detail, we proved that there is no pre-period,
, and the primitive period
of length
consists of
5, resp. 9, metabelian vertices with
, resp.
, and
8, resp. 16, non-metabelian vertices with
, resp.
,
(
children of
, and
children of
with depth 1)
together 38 vertices (14 of them metabelian) in branch
, and
9, resp. 10, metabelian vertices with
, resp.
, and
16, resp. 16, non-metabelian vertices with
, resp.
,
(
children of
, and
children of
, both with depth 1)
together 51 vertices (19 of them metabelian) in branch
.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung ( [7] , Thm. 5.1.16, pp. 178-179, and the fourth Figure, e ≥ 5,
, on the double page between pp. 191-192). The tree
corresponds to the infinite metabelian pro-3 group
in ( [6] , Cnj. 15 (b), p. 116).
The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [1] and [2] . Here, the strict periodicity was confirmed by computation up to branch
and undoubtedly sets in at
.
Theorem 11.3. (Graph theoretic and algebraic invariants.)
The coclass-4 tree
of 3-groups G with coclass
which arises from the metabelian root
has the following abstract graph theoretic properties.
1) The branches
,
, are purely periodic with primitive period
of length
.
2) The cardinalities of the periodic branches are
and
.
3) Depth, width, and information content of the tree are given by
(11.3)
The algebraic invariants of the vertices forming the primitive period
of the tree are given in Table 3. The six leading branches
are drawn in Figure 7.
Proof. (of Theorem 11.3) Since every mainline vertex
of the tree
has several capable children,
, but every capable vertex v of depth 1 has only terminal children,
, according to Proposition 11.4, the depth of the tree is
. In this case, the cardinality of a branch
is the sum of the number
of immediate descendants of the branch root
and the numbers
of terminal children of capable vertices
of depth 1 with
(excluding the next mainline vertex
), according to Formula (5.10), that is,
![]()
Table 3. Data for 3-groups G with
of the coclass tree
.
![]()
Figure 7. The first coclass-4 tree
with mainline of type d.19*.
![]()
Applied to the primitive period, this yields
,
.According to Formula (5.14), the width of the tree is the maximum of all sums of the shape
![]()
taken over all branch roots
with logarithmic orders
.Applied to
,
,and
,this yields
.
Finally, we have
.
Corollary 11.2. (Actions and relation ranks.) The algebraic invariants of the vertices of the structured coclass-4 tree
are listed in Table 3. In par- ticular:
1) There are no groups with
-action.
2) Two distinguished terminal metabelian vertices of depth 2 with even class and type H.4, all terminal vertices of depth 1 with odd class, and the mainline vertices with even class, possess an RI-action.
3) The relation rank is given by
for the mainline vertices
with
, and the capable vertices
of depth 1 with
, and
otherwise.
Proof. (of Corollary 11.2) The existence of an RI-action on G has been checked by means of an algorithm involving the p-covering group of G, written for MAGMA [17] . The other claims follow immediately from Table 3, continued indefinitely with the aid of the periodicity in Prop. 11.4.
Theorem 11.4. (Strict isomorphism of the two trees.)
Viewed as an algebraically structured infinite digraph, the second coclass-4 tree
with mainline of type d.19* in Figure 8 is strictly isomorphic to the first coclass-4 tree
with mainline of type d.19* in Figure 7. Only the presentations of corresponding vertices are different, but they share common algebraic invariants.
Proof. (Proof of Theorem 11.3 and Theorem 11.4) The claims have been verified with the aid of MAGMA [17] for all vertices v with logarithmic orders
. Pure periodicity of branches with primitive length 2 sets in from the very beginning with
. There is no pre-period. Thus, the claims for all vertices v with logarithmic orders
are a consequence of the virtual periodicity theorems by du Sautoy in ( [1] , Thm. 1.11, p. 68, and Thm. 8.3, p. 103) and by Eick and Leedham-Green in ( [2] , Thm. 6, p. 277, Thm. 9, p. 278, and Thm. 29, p. 287], without the need of pruning the depth, which is bounded uniformly by 2.
11.3. The Unique Mainline of Type d.23* for Even Coclass r ≥ 4
Proposition 11.5. (A special nearly strict isomorphism.)
Viewed as an algebraically structured infinite digraph, the unique coclass-4 tree
with mainline of type d.23* in Figure 9 is almost strictly isomorphic to the first coclass-4 tree
with mainline of type d.19* in Figure 7, and thus also to the second coclass-4 tree
with mainline of type d.19* in Figure 8. Only the
![]()
Figure 8. The second coclass-4 tree
with mainline of type d.19*.
presentations of corresponding vertices are different, but they share common algebraic invariants, with the transfer kernel types as single exception: the nearly strict isomorphism of directed trees maps
,
,
, and F.12 either remains fixed or
.
Proof. This follows immediately from comparing Table 4 with Table 3, and using periodicity.
Proposition 11.6. (Periodicity and descendant numbers.)
The branches
,
, of the unique coclass-4 tree
with mainline vertices of transfer kernel type d.23*,
![]()
Figure 9. The unique coclass-4 tree
with mainline of type d.23*.
, are purely periodic with primitive length
and without pre-period,
, that is,
are isomorphic as digraphs, for all
.
The graph theoretic structure of the tree is determined uniquely by the numbers
of immediate descendants and
of capable immediate descendants of the mainline vertices
with logarithmic order
and of capable vertices v with depth 1 and
:
![]()
Table 4. Data for 3-groups G with
of the coclass tree
.
for mainline vertices
with odd logarithmic order
,
for mainline vertices
with even logarithmic order
,
for the capable vertex v of depth 1 and even logarithmic order
,
for two capable vertices v of depth 1 and odd logarithmic order
.
Proof. This is a consequence of Proposition 11.5 together with Proposition 11.4. The tree
corresponds to the infinite metabelian pro-3 group
in ( [6] , Cnj. 15 (b), p. 116).
Theorem 11.5. (Graph theoretic and algebraic invariants.)
The coclass-4 tree
of 3-groups G with coclass
which arises from the metabelian root
has the following abstract graph theoretic properties.
1) The branches
,
, are purely periodic with primitive period
of length
.
2) The cardinalities of the periodic branches are
and
.
3) Depth, width, and information content of the tree are given by
(11.4)
The algebraic invariants of the vertices forming the primitive period
of the tree are given in Table 4. The six leading branches
are drawn in Figure 9.
Proof. Since the tree
is isomorphic to the tree
as an abstract digraph, the proof literally coincides with the proof of Theorem 11.3.
Corollary 11.3. (Actions and relation ranks.) The algebraic invariants of the vertices of the structured coclass-4 tree
are listed in Table 4. In particular:
1) There are no groups with
-action.
2) Two distinguished terminal metabelian vertices of depth 2 with even class and type G.16, all terminal vertices of depth 1 with odd class, and the mainline vertices with even class, possess an RI-action.
3) The relation rank is given by
for the mainline vertices
with
, and the capable vertices
of depth 1 with
, and
otherwise.
11.4. Two Mainlines of Type d.25* for Even Coclass r ≥ 4
Proposition 11.7. (Periodicity and descendant numbers.)
The branches
,
, of the first coclass-4 tree
with mainline vertices of transfer kernel type d.25*,
, are purely periodic with primitive length
and without pre- period,
, that is,
are isomorphic as graphs, for all
.
The structure of the tree is determined uniquely by the numbers
of immediate descendants and
of capable immediate descendants for mainline vertices and for capable vertices of depth 1:
for mainline vertices
of odd logarithmic order
,
for mainline vertices
of even logarithmic order
,
for a capable vertex v of depth 1 and even logarithmic order
,
for two capable vertices v of depth 1 and odd logarithmic order
.
Theorem 11.6. (Graph theoretic and algebraic invariants.)
The coclass-4 tree
of 3-groups G with coclass
which arises from the metabelian root
has the following abstract graph theoretic properties.
1) The branches
,
, are purely periodic with primitive period
of length
.
2) The cardinalities of the periodic branches are
and
.
3) Depth, width, and information content of the tree are given by
(11.5)
The algebraic invariants of the vertices forming the primitive period
of the tree are presented in Table 5. The leading six branches
are drawn in Figure 10.
![]()
Table 5. Data for 3-groups G with
of the coclass tree
.
![]()
Figure 10. The first coclass-4 tree
with mainline of type d.25*.
Corollary 11.4. (Actions and relation ranks.)
The algebraic invariants of the vertices of the structured coclass-4 tree
are listed in Table 5. In particular:
1) All mainline vertices, two capable metabelian vertices of depth 1 with odd class and type G.19, two distinguished terminal metabelian vertices of depth 2 with even class and type G.19, and two distinguished terminal non-metabelian vertices of depth 1 with odd class and type d.25 possess a
-action.
2) Two distinguished terminal metabelian vertices of depth 2 with even class and type G.19, all terminal vertices of depth 1 with odd class, and the mainline vertices with even class, possess an RI-action.
3) The relation rank is given by
for the mainline vertices
with
, and the capable vertices
of depth 1 with
, and
otherwise.
Proof. (of Proposition 11.7, Theorem 11.6, and Corollary 11.4) The proofs are very similar to those of Proposition 11.4, Theorem 11.3, and Corollary 11.2. The differences are only the concrete numerical values of the invariants involved in the calculations:
,
,
,
and
.
In detail, we proved that there is no pre-period,
, and the primitive period
of length
consists of
4, resp. 6, metabelian vertices with
, resp.
, and
5, resp. 9, non-metabelian vertices with
, resp.
,
(
children of
, and
children of
with depth 1)
together 24 vertices (10 of them metabelian) in branch
, and
6, resp. 8, metabelian vertices with
, resp.
, and
9, resp. 10, non-metabelian vertices with
, resp.
,
(
children of
, and
children of
, both with depth 1)
together 33 vertices (14 of them metabelian) in branch
.
The tree
corresponds to the infinite metabelian pro-3 group
in ( [6] , Cnj. 15 (b), p. 116).
Theorem 11.7. (Strict isomorphism of the two trees.)
Viewed as an algebraically structured infinite digraph, the second coclass-4 tree
with mainline of type d.25* in Figure 11 is strictly isomorphic to the first coclass-4 tree
with mainline of type d.25* in Figure 10. Only the presentations of corresponding vertices are different, but they share common algebraic invariants.
Proof. (Proof of Thm. 11.6 and Thm 11.7.) The claims have been verified with the aid of MAGMA [17] for all vertices V with logarithmic orders
. Pure periodicity of branches sets in with
. Thus, the claims for all vertices V with logarithmic orders
are a consequence of the periodicity theorems by du Sautoy in [1] and by Eick and Leedham-Green in [2] , without the need of pruning the depth, which is bounded uniformly by 2.
![]()
Figure 11. The second coclass tree
with mainline of type d.25*.
12. Sporadic and Periodic 3-Groups G of Odd Coclass ![]()
Although formulated for the particular coclass
, all results on sporadic and periodic groups in this section are valid for any odd coclass
. The exemplary (co-periodic) sporadic part
of the coclass forest
is presented in the following Proposition 12.1.
Proposition 12.1. The sporadic part
of the coclass-5 forest
consists of
• 7
isolated metabelian vertices with types F.7, F.11, F.12, F.13,
• 4
metabelian roots of finite trees with types G.16, G.19, H.4, together with their 24 metabelian and 36 non-metabelian children, all with depth
,
• 34
isolated vertices with
and types d.19, d.23, d.25,
• 89 isolated vertices with
and type b.10,
• 13 capable vertices with
and type b.10,
whose children do not belong to
, by definition.
The action flag of all vertices is
, and consequently none of them has an RI- or
-action.
Together with the 4 metabelian roots of coclass-5 trees, the
vertices of depth
are exactly the
children of step size
of
, and the
capable vertices among them correspond to the invariant
of
.
Figure 12 sketches an outline of the metabelian skeleton of the coclass forest
in its top region. The vertices
and
, with the crucial bifurcation from
to
, belong to the infinite main trunk (§10).
Theorem 12.1. The coclass-r forest
with any odd
is the disjoint union of its finite sporadic part
with total information content
(12.1)
and
infinite coclass-r trees
with roots
, where
for
. The algebraic invariants for groups with centre
, and in cumulative form for
, are given for
in Table 6, where the parent vertex
on the maintrunk is also included, but the 136 non-metabelian top vertices of depth
are excluded.
Proof. (of Proposition 12.1 and Theorem 12.1) We have computed the sporadic parts
of coclass forests
with odd
up to
by means of MAGMA [17] . They all share a common graph theoretic structure with
. The forest
contains 4 roots of coclass trees with metabelian mainlines (a unique root
of type b and three roots
of type d), namely
,
,
,
,
which give rise to the periodic part of
, and 35 sporadic metabelian groups of type F, G or H. Among the groups of the sporadic part
, there are 7 isolated metabelian vertices with type F, and 4 metabelian roots of finite trees with type G or H and tree depth 1. Among the ![]()
![]()
Figure 12. Metabelian interface between the coclass forests
and
.
children, there are
metabelian, and
have derived length 3. The latter are omitted in the forest diagram, Figure 12. Additionally,
contains 136 non-metabelian top vertices, which gives a total information content
of
representatives.
The metabelian skeleton consists of
vertices. The results for metabelian groups are in accordance with the third tree diagram
,
, in ( [7] , third page between pp. 191-192). The metabelian groups in Table 6 correspond to the representatives of isomorphism classes in ( [28] , pp. 34-35).
12.1. The Unique Mainline of Type b.10* for Odd Coclass r ≥ 5
Proposition 12.2. (Periodicity and descendant numbers.)
![]()
Table 6. Data for sporadic 3-groups G with
of the forest
.
The branches
,
, of the coclass-5 tree
with mainline vertices of transfer kernel type b.10*,
, are periodic with pre-period length
and with primitive period length
, that is,
are isomorphic as digraphs, for all
.
The graph theoretic structure of the tree is determined uniquely by the numbers
of immediate descendants and
of capable immediate descen- dants of the mainline vertices
with logarithmic order
:
for the root
with
,
for all mainline vertices
with even logarithmic order
,
for all mainline vertices
with odd logarithmic order
.
Proof. (of Proposition 12.2) The statements concerning the numbers
of immediate descendants of the mainline vertices
with
have been obtained by direct computation with the p-group generation algorithm [12] [13] [14] in MAGMA [17] . In detail, we proved that there are
6, resp. 6, metabelian vertices with bicyclic centre
, resp. cyclic centre
, and
9, resp. 9, non-metabelian vertices with
, resp.
,
together 30 vertices (12 of them metabelian) in the pre-periodic branch
,
and the primitive period
of length
consists of
4, resp. 6, metabelian vertices with
, resp.
, and
5, resp. 9, non-metabelian vertices with
, resp.
,
together 24 vertices (10 of them metabelian) in branch
, and
6, resp. 9, metabelian vertices with
, resp.
, and
9, resp. 16, non-metabelian vertices with
, resp.
,
together 40 vertices (15 of them metabelian) in branch
.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung ( [7] , Thm. 5.1.16, pp. 178-179, and the third Figure, e ≥ 4, e
0 (mod 2), on the third page between pp. 191-192]). The tree
corresponds to the infinite metabelian pro-3 group
in ( [6] , Cnj. 15 (a), p. 116).
The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [1] and [2] . Here, the strict periodicity was confirmed by computation up to branch
and clearly sets in at
.
Theorem 12.2. (Graph theoretic and algebraic invariants.)
The coclass-5 tree
of finite 3-groups G with coclass
which arises from the metabelian root
has the following graph theoretic properties.
1) The pre-period
of length
is irregular.
2) The cardinality of the irregular branch is
.
3) The branches
,
, are periodic with primitive period
of length
.
4) The cardinalities of the regular branches are
and
.
5) Depth, width, and information content of the tree are given by
(12.2)
The algebraic invariants of the groups represented by vertices forming the pre-period
and the primitive period
of the tree are given in Table 7. The six leading branches
are drawn in Figure 13.
Remark 12.1. The algebraic information in Table 7 is visualized in Figure 13. By periodic continuation, the figure shows more branches than the table but less details concerning the exact order
of the automorphism group.
Proof. (of Theorem 12.2) According to Proposition 12.2, the logarithmic order of the tree root, respectively of the periodic root, is
, respectively
.
Since
for all mainline vertices
with
, according to Proposition 12.2, the unique capable child of
is
, and each branch has
![]()
Table 7. Data for 3-groups G with
of the coclass tree
.
![]()
Figure 13. The unique coclass-5 tree
with mainline of type b.10*.
depth
, for
. Consequently, the tree is also of depth
.
With the aid of Formula (5.9) in Theorem 5.1, the claims (2) and (4) are consequences of Proposition 12.2:
,
,
and
.
According to Formula (5.13) in Corollary 5.1, where n runs from
to
, the tree width is the maximum
of the expressions
,
, and
.
The information content of the tree is given by Formula (5.17) in the Definition 5.3:
.
The algebraic invariants in Table 7, that is, depth
, derived length
, abelian type invariants of the centre
, relation rank
, nuclear rank
, abelian quotient invariants
of the first maximal subgroup, respectively
of the commutator subgroup, transfer kernel type
, action flag
, and the factorized order
of the automorphism group have been computed by means of program scripts written for MAGMA [17] .
Each group is characterized by the parameters of the normalized repre- sentative
of its isomorphism class, according to Formula (9.2), and by its identifier
in the SmallGroups Database [19] .
The column with header # contains the number of groups with identical invariants (except the presentation), for each row.
Corollary 12.1. (Actions and relation ranks.) The algebraic invariants of the vertices of the structured coclass-5 tree
are listed in Table 7. In par- ticular:
1) The groups with
-action are all mainline vertices
,
, the two terminal vertices
with even
, two terminal non-metabelian vertices with
and even
, three pre-periodic terminal vertices
, and two pre-periodic terminal non-metabelian vertices with
and
.
2) With respect to the kernel types, all mainline groups of type b.10*,
, the two leaves of type d.25,
, with every even logari- thmic order, two distinguished non-metabelian leaves of type b.10 with every even logarithmic order, and five pre-periodic leaves of type b.10 with
possess a
-action.
3) All terminal vertices of depth 1 with odd class and the mainline vertices with even class possess an RI-action.
4) The relation rank is given by
for the mainline vertices
,
, and
otherwise.
Proof. (of Corollary 12.1) The existence of an RI-action on G has been checked by means of an algorithm involving the p-covering group of G, written for MAGMA [17] . The other claims follow immediately from Table 7, continued indefinitely with the aid of the periodicity in Proposition 12.2.
12.2. The Unique Mainline of Type d.19* for Odd Coclass r ≥ 5
Proposition 12.3. (Periodicity and descendant numbers.)
The branches
,
, of the unique coclass-5 tree
with mainline vertices of transfer kernel type d.19*,
, are purely periodic with primitive length
and without pre-period,
, that is,
are isomorphic as structured digraphs, for all
.
The graph theoretic structure of the tree is determined uniquely by the numbers
of immediate descendants and
of capable immediate descendants for mainline vertices
with logarithmic order
and for capable vertices v with depth 1 and
:
for all mainline vertices
of any logarithmic order
,
for two capable vertices v of depth 1 and any logarithmic order
.
Proof. (of Proposition 12.3) The statements concerning the numbers
of immediate descendants of the mainline vertices
with
, and
of vertices with depth
and logarithmic order
, have been obtained by direct computation with the p-group generation algorithm [12] [13] [14] in MAGMA [17] . In detail, we proved that there is no pre-period,
, and the primitive period
of length
consists of:
9, resp. 18, metabelian vertices with
, resp.
, and
16, resp. 32, non-metabelian vertices with
, resp.
,
(i.e.
children of
, and
children of
, both with depth 1)
together 75 vertices (27 of them metabelian) in branch
.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung ( [7] , Thm. 5.1.16, pp. 178-179, and the third Figure, e ≥ 4,
, on the third page between pp. 191-192) The tree
corresponds to the infinite metabelian pro-3 group
in ( [6] , Cnj. 15 (a), p. 116).
The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [1] and [2] . Here, the strict periodicity was confirmed by computation up to branch
and certainly sets in at
.
Theorem 12.3. (Graph theoretic and algebraic invariants.)
The coclass-5 tree
of 3-groups G with coclass
which arises from the metabelian root
has the following abstract graph theoretic properties.
1) The branches
,
, are purely periodic with primitive period
of length
.
2) The cardinality of the periodic branch is
.
3) Depth, width, and information content of the tree are given by
(12.3)
The algebraic invariants of the vertices forming the root and the primitive period
of the tree are presented in Table 8. The leading six branches
are drawn in Figure 14.
Proof. (Proof of Theorem 12.3) Since every mainline vertex
of the tree
has three capable children,
, but every capable vertex v of depth 1 has only terminal children,
, according to Proposition 12.3, the depth of the tree is
. In this case, the cardinality of a branch
is
![]()
Figure 14. The unique coclass-5 tree
with mainline of type d.19*.
![]()
Table 8. Data for 3-groups G with
of the coclass tree
.
the sum of the number
of immediate descendants of the branch root
and the numbers
of terminal children of capable vertices
of depth 1 with
(excluding the next mainline vertex
), according to Formula (5.10), that is,
![]()
Applied to the primitive period, this yields
. According to Formula (30), the width of the tree is the maximum of all sums of the shape
![]()
taken over all branch roots
with logarithmic orders
. Applied to
,
, and
, this yields
.
Finally, we have
.
Corollary 12.2. (Actions and relation ranks.)
The algebraic invariants of the vertices of the structured coclass-5 tree
are listed in Table 8. In particular:
1) There are no groups with GI-action, let alone with RI- or
-action.
2) The relation rank is given by
for the mainline vertices
with
, and the capable vertices
of depth 1 with
, and
otherwise.
Proof. (of Corollary 12.2) The existence of an RI-action on G has been checked by means of an algorithm involving the p-covering group of G, written for MAGMA [17] . The other claims follow immediately from Table 8, continued indefinitely with the aid of the periodicity in Proposition 12.3.
12.3. The Unique Mainline of Type d.23* for Odd Coclass r ≥ 5
Proposition 12.4. (Periodicity and descendant numbers.)
The branches
,
, of the unique coclass-5 tree
with mainline vertices of transfer kernel type d.23*,
, are purely periodic with primitive length
and without pre-period,
, that is,
are isomorphic as structured digraphs, for all
.
The graph theoretic structure of the tree is determined uniquely by the numbers
of immediate descendants and
of capable immediate descendants for mainline vertices
with logarithmic order
and for capable vertices v with depth 1 and
:
for all mainline vertices
of any logarithmic order
,
for the capable vertex v of depth 1 and any logarithmic order
.
Proof. (of Proposition 12.4) The statements concerning the numbers
of immediate descendants of the mainline vertices
with
, and
of vertices with depth
and logarithmic order
, have been obtained by direct computation with the p-group generation algorithm [12] [13] [14] in MAGMA [17] . In detail, we proved that there is no pre-period,
, and the primitive period
of length
consists of
6, resp. 9, metabelian vertices with
, resp.
, and
9, resp. 16, non-metabelian vertices with
, resp.
,
(i.e.
children of
, and
) children of
with depth 1)
together 40 vertices (15 of them metabelian) in branch
.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung ( [7] , Thm. 5.1.16, pp. 178-179, and the third Figure, e ≥ 4,
, on the third page between pp. 191-192). The tree
corresponds to the infinite metabelian pro-3 group
in ( [6] , Cnj. 15 (a), p. 116).
The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [1] and [2] . Here, the strict periodicity was confirmed by computation up to branch
and certainly sets in at
.
Theorem 12.4. (Graph theoretic and algebraic invariants.)
The coclass-5 tree
of 3-groups G with coclass
arises from the metabelian root
and has the following abstract graph theoretic properties.
1) The branches
,
, are purely periodic with primitive period
of length
.
2) The cardinality of the periodic branch is
.
3) Depth, width, and information content of the tree are given by
(12.4)
The algebraic invariants of the vertices forming the root and the primitive period
of the tree are presented in Table 9. The leading six branches
are drawn in Figure 15.
![]()
Figure 15. The unique coclass-5 tree
with mainline of type d.23*.
![]()
Table 9. Data for 3-groups G with
of the coclass tree
.
Proof. (Proof of Theorem 12.4) Since every mainline vertex
of the tree
has two capable children,
, but every capable vertex v of depth 1 has only terminal children,
, according to Proposition 12.4, the depth of the tree is
. In this case, the cardinality of a branch
is the sum of the number
of immediate descendants of the branch root
and the numbers
of terminal children of capable vertices
of depth 1,
(
the next mainline vertex must be omitted), according to Formula (5.10),
![]()
Applied to the primitive period, this yields
. According to Formula (5.14), the width of the tree is the maximum of all sums of the shape
![]()
taken over all branch roots
with logarithmic orders
. Applied to
,
, and
, this yields
. Finally, we have
.
Corollary 12.3. (Actions and relation ranks.) The algebraic invariants of the vertices of the structured coclass-5 tree
are listed in Table 9. In par- ticular:
1) There are no groups with GI-action, let alone with RI- or
-action.
2) The relation rank is given by
for the mainline vertices
with
, and the capable vertices
of depth 1 with
, and
otherwise.
Proof. (of Corollary 12.3) The existence of an RI-action on G has been checked by means of an algorithm involving the p-covering group of G, written for MAGMA [17] . The other claims follow immediately from Table 9, continued indefinitely with the aid of the periodicity in Prop. 12.4.
12.4. The Unique Mainline of Type d.25* for Odd Coclass r ≥ 5
Proposition 12.5. (Periodicity and descendant numbers.)
The branches
,
, of the unique coclass-5 tree
with mainline vertices of transfer kernel type d.25*,
, are purely periodic with primitive length
and without pre-period,
, that is,
are isomorphic as structured digraphs, for all
.
The graph theoretic structure of the tree is determined uniquely by the numbers
of immediate descendants and
of capable immediate descendants for mainline vertices
with logarithmic order
and for capable vertices v with depth 1 and
:
for all mainline vertices
of odd logarithmic order
,
for all mainline vertices
of even logarithmic order
,
for two capable vertices v of depth 1 and even logarithmic order
,
for the capable vertex v of depth 1 and odd logarithmic order
.
Proof. (of Proposition 12.5) The statements concerning the numbers
of immediate descendants of the mainline vertices
with
, and
of vertices with depth
and logarithmic order
, have been obtained by direct computation with the p-group generation algori- thm [12] [13] [14] in MAGMA [17] . In detail, we proved that there is no pre-period,
, and the primitive period
of length
consists of
6, resp. 12, metabelian vertices with
, resp.
, and
9, resp. 18, non-metabelian vertices with
, resp.
,
(i.e.
children of
, and
children of
, both with depth 1)
together 45 vertices (18 of them metabelian) in branch
, and
6, resp. 9, metabelian vertices with
, resp.
, and
9, resp. 16, non-metabelian vertices with
, resp.
,
(i.e.
children of
, and
children of
with depth 1)
together 40 vertices (15 of them metabelian) in branch
.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung ( [7] , Thm. 5.1.16, pp. 178-179, and the third Figure, e ≥ 4,
, on the third page between pp. 191-192]. The tree
corresponds to the infinite metabelian pro-3 group
in ( [6] , Cnj. 15 (a), p. 116).
The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [1] and [2] . Here, the strict periodicity was confirmed by computation up to branch
and certainly sets in at
.
Theorem 12.5. (Graph theoretic and algebraic invariants.)
The coclass-5 tree
of 3-groups G with coclass
which arises from the metabelian root
has the following abstract graph theoretic properties.
1) The branches
,
, are purely periodic with primitive period
of length
.
2) The cardinalities of the periodic branches are
and
.
3) Depth, width, and information content of the tree are given by
(12.5)
The algebraic invariants of the vertices forming the root and the primitive period
of the tree are presented in Table 10. The leading six branches
are drawn in Figure 16.
Proof. (Proof of Theorem 12.5) Since every mainline vertex
of the tree
has several capable children,
, but every capable vertex v of depth 1 has only terminal children,
, according to Proposition 12.5, the depth of the tree is
. In this case, the cardinality of a branch
is the sum of the number
of immediate descendants of the branch root
and the numbers
of terminal children of capable vertices
of depth 1,
(where
is the next mainline vertex and must be discouraged), according to Formula (5.10),
Applied to the primitive period, this yields
and
. According to Formula (5.14), the width of the tree is the maximum of all sums of the shape
taken over all branch roots
with logarithmic orders
. Applied to
,
, and
, this yields
. Finally, we have
.Corollary 12.4. (Actions and relation ranks.)
![]()
Figure 16. The unique coclass-5 tree
with mainline of type d.25*.
The algebraic invariants of the vertices of the structured coclass-5 tree
are listed in Table 10. In particular:
1) There are no groups with GI-action, let alone with RI- or
-action.
2) The relation rank is given by
for the mainline vertices
with
, and the capable vertices
of depth 1 with
, and
otherwise.
![]()
Table 10. Data for 3-groups G with
of the coclass tree
.
Proof. (of Corollary 12.4) The existence of an RI-action on G has been checked by means of an algorithm involving the p-covering group of G, written for MAGMA [17] . The other claims follow immediately from Table 10, continued indefinitely with the aid of the periodicity in Proposition 12.5.
13. The Forest of 3-Groups with Coclass 1
The coclass forests
and
, and even their metabelian skeletons, sporadic parts, and individual coclass trees, do not reveal any isomorphism to higher coclass forests
, with
, or parts of them. The metabelian skeleton of the coclass forest
is isomorphic to the metabelian skeleton of any coclass forest
, with odd
, according to Nebelung [7] , but neither its sporadic part nor its coclass trees are isomorphic to the corres- ponding components of other coclass forests.
Whereas the complexity of the pre-periodic forests
and
is very high, the simplest forest
can be described easily. In particular, the coclass-1 forest
coincides with its unique coclass tree arising from the abelian root
. Its sporadic part
is void.
The Unique Mainline of Type a.1* for Coclass r = 1
Proposition 13.1. (Periodicity and descendant numbers.)
The branches
,
, of the coclass-1 tree
with mainline vertices of transfer kernel type a.1*,
, are periodic with pre-period length
and with primitive period length
, that is,
are isomorphic as digraphs, for all
.
The graph theoretic structure of the tree is determined uniquely by the numbers
of immediate descendants and
of capable immediate descendants of the mainline vertices
with logarithmic order
:
for the root
with
,
for the mainline vertex
with
,
for mainline vertices
with even logarithmic order
,
for mainline vertices
with odd logarithmic order
.
Proof. (of Proposition 13.1) The statements concerning the numbers
of immediate descendants of the mainline vertices
with
are due to Blackburn ( [9] , Thm. 4.2 and Thm 4.3, p. 88), who distinguishes the groups according to their defect of commutativity
, which is defined by
in terms of the lower central series
, nilpotency class
, and the two-step centralizer
of G.
In detail, Blackburn proved that there are
4 vertices v with defect
in the pre-periodic branch
,
and the primitive period
of length
consists of
3 vertices v with defect
, and 3 vertices v with defect
,
together 6 vertices in branch
, and
4 vertices v with defect
, and 3 vertices v with defect
,
together 7 vertices in branch
. All vertices of the tree are metabelian.
The results were reproduced and supplemented with
by Nebe- lung ( [7] , Thm. 5.1.17, pp. 179-180), and have been verified by ourselves independently by direct computation with MAGMA [17] , where the p-group generation algorithm by Newman and O'Brien [12] [13] [14] is implemented.
Accordingly, the pre-period
of length
consists of
2 vertices v with defect
in branch
, and
4 vertices v with defect
in branch
.
The claim of the virtual periodicity of branches has been proved generally for any coclass tree by du Sautoy [1] , and independently by Eick and Leedham-Green [2] . Here, the strict periodicity is also a consequence of Blackburn’s results, and has been tested up to
computationally.
Theorem 13.1. (Graph theoretic and algebraic invariants.)
The coclass-1 tree
of all finite 3-groups
with coclass
arises from the abelian root
and has the following graph theoretic properties.
1) The pre-period
of length
is irregular.
2) The cardinalities of the irregular branches are
and
.
3) The branches
,
, are periodic with primitive period
of length
.
4) The cardinalities of the regular branches are
and
.
5) Depth, width, and information content of the tree are given by
(13.1)
The algebraic invariants of the groups represented by vertices forming the pre-period
and the primitive period
of the tree are given in Table 11. The leading eight branches
are drawn in Figure 17. All vertices of the tree are metabelian.
![]()
Table 11. Data for 3-groups G with
of the coclass tree
.
![]()
Figure 17. The unique coclass-1 tree
with mainline of type a.1*.
Proof. (of Theorem 13.1) According to Proposition 13.1, the logarithmic order of the tree root, respectively of the periodic root, is
, respectively
.
Since
for all mainline vertices
with
, according to Proposition 13.1, the unique capable child of
is
, and each branch has depth
, for
. Consequently, the tree is also of depth
.
With the aid of Formula (5.9) in Theorem 5.1, the claims (2) and (4) are consequences of Proposition 13.1:
,
,
, and
.
According to Formula (5.13) in Corollary 5.1, where n runs from
to
, the tree width is the maximum
of the expressions
,
,
, and
.
The information content of the tree is given by Formula (5.17) in the Definition 5.3:
![]()
The algebraic invariants in Table 11, that is, defect of commutativity k, depth
, derived length
, abelian type invariants of the centre
, relation rank
, nuclear rank
, abelian quotient invariants
of the first maximal subgroup, respectively
of the commutator subgroup, transfer kernel type
, action flag
, and the factorized order
of the automorphism group have been computed by means of program scripts written for MAGMA [17] .
Each group is characterized by the parameters of the normalized repre- sentative
of its isomorphism class, according to Formula (9.1), and by its identifier
in the SmallGroups Database [19] .
The column with header # contains the number of groups with identical invariants (except the presentation and identifier), for each row.
Corollary 13.1. (Actions and relation ranks.)
The algebraic invariants of the vertices of the structured coclass-1 tree
with abelian root
, which is drawn in Figure 17, are listed in Table 11. In particular:
1) The groups with
-action are the root R, all mainline vertices
,
, and the terminal vertices
with even logarithmic order
.
2) With respect to the transfer kernel types, all mainline groups of type a.1*,
, and the leaves of type a.3,
, with odd class
, possess a
-action.
3) The relation rank is given by
for the mainline vertices
,
,
for the terminal extraspecial group
, and
otherwise.
4) All terminal vertices with odd class, and the mainline vertices with even class, possess an RI-action. The terminal vertices with odd class are Schur + 1
-groups [29] [30] .
Proof. (of Corollary 13.1) The existence of an RI-action on G has been checked by means of an algorithm involving the p-covering group of G, written for MAGMA [17] . The other claims follow immediately from Table 11, continued indefinitely with the aid of the periodicity in Proposition 13.1. A Schur + 1
-group has an RI-action and relation rank
[29] [30] .
14. Conclusions
In the core Sections 11 and 12 of this paper, we have elaborated our long desired proof that the pruned tree of all finite 3-groups with elementary bicyclic commutator quotient, which do not arise as descendants of non-metabelian groups, can be described with a finite amount of data.
Theorem 14.1. (Main Theorem on the Finite Information Content.)
The total information content of the coclass forest
is given by
(14.1)
Proof. The total information content of a coclass forest is the sum of the cardinality of its sporadic part
and the information contents of its pairwise non-isomorphic coclass trees ![]()
![]()
Due to the exceptional complexity of the pre-periodic forests
and
, their information contents are unknown up to now. However, since they are certainly finite, this does not obfuscate our clear and beautiful results concerning the infinitely many co-periodic forests
with
, which can be reduced to the finite information content of the primitive co-period
.
Acknowledgements
We gratefully acknowledge that our research was supported by the Austrian Science Fund (FWF): Project P 26008-N25. Indebtedness is expressed to the anonymous referees for valuable suggestions concerning the readability and, in particular, for drawing our attention to the paper [6] .
Supported
Research supported by the Austrian Science Fund (FWF): P 26008-N25.