1. Introduction
In §§2 - 11, we present an exposition of facts concerning the mathematical structure which forms the central idea of this article: descendant trees of finite p-groups. Their computational construction is recalled in §§12 - 20 on the p-group generation algorithm. Recently periodic patterns have been discovered in descendant trees with promising arithmetical applications form the topic of the final §21 and the coronation of the entire work.
2. Thestructure: Descendant trees
In mathematics, specifically group theory, a descendant tree is a hierarchical structure for visualizing parent- descendant relations (§§4 and 6) between isomorphism classes of finite groups of prime power order
, for a fixed prime number
and varying integer exponents
. Such groups are briefly called finite p-groups. The vertices of a descendant tree are isomorphism classes of finite
-groups.
Additionally to their order
, finite p-groups possess two further related invariants, the nilpotency class
and the coclass
(§§5 and 8). It turned out that descendant trees of a particular kind, the so-called pruned coclass trees whose infinitely many vertices share a common coclass
, reveal a repeating finite pattern (§7). These two crucial properties of finiteness and periodicity, which have been proved independently by du Sautoy [1] and by Eick and Leedham-Green [2] , admit a characterization of all members of the tree by finitely many parametrized presentations (§§10 and 21). Consequently, descendant trees play a fundamental role in the classification of finite p-groups. By means of kernels and targets of Artin transfer homomorphisms [3] , de- scendant trees can be endowed with additional structure [4] -[6] , which recently turned out to be decisive for ari- thmetical applications in class field theory, in particular, for determining the exact length of p-class towers [7] .
An important question is how the descendant tree
can actually be constructed for an assigned starting group which is taken as the root
of the tree. Sections §§13 - 19 are devoted to recall a minimum of the necessary background concerning the p-group generation algorithm by Newman [8] and O’Brien [9] [10] , which is a recursive process for constructing the descendant tree of a foregiven finite p-group playing the role of the tree root. This algorithm is now implemented in the ANUPQ-package [11] of the computational algebra systems GAP [12] and MAGMA [13] .
As a final highlight in §21, whose formulation requires an understanding of all the preceding sections, this article concludes with brand-new discoveries of an unknown, and up to now unproved, kind of repeating infinite patterns called periodic bifurcations, which appeared in extensive computational constructions of descendant trees of certain finite 2-groups, resp. 3-groups, G with abelianization
of type (2,2,2), resp. (3,3), and have immediate applications in algebraic number theory and class field theory.
3. Historical remarks onbifurcation
Since computer aided classifications of finite
-groups go back to 1975, fourty years ago, there arises the question why periodic bifurcations did not show up in the earlier literature already. At the first sight, this fact seems incomprehensible, because the smallest two 3-groups which reveal the phenomenon of periodic bifurcations with modest complexity were well known to both, Ascione, Havas and Leedham-Green [14] and Nebelung [15] . Their SmallGroups identifiers are
and
(see §9 and [16] [17] ). Due to the lack of systematic identifiers in 1977, they were called the non-CF groups Q and U in ([14] , Table 1, p. 265, and Table 2, p. 266), since their lower central series
has a non-cyclic factor
of type (3,3). Similarly, there was no SmallGroups Database yet in 1989, whence the two groups were designated by
and
in ([15] , Satz 6.14, p. 208).
So Ascione and Nebelung were both standing in front of the door to a realm of uncharted waters. The reason why they did not enter this door was the sharp definition of their project targets. A bifurcation is the special case of a 2-fold multifurcation (§8): At a vertex
of coclass
with nuclear rank
, the de- scendant tree
forks into a regular component of the same coclass
and an irregular component of the next coclass
.
Ascione’s thesis subject [18] [19] in 1979 was to investigate two-generated 3-groups
of second maximal class, that is, of coclass
. Consequently, she studied the regular tree
for
and did not touch the irregular component
whose members are not of second maximal class.
The goal of Nebelung’s dissertation [15] in 1989 was the classification of metabelian 3-groups
with
of type
. Therefore she focused on the metabelian skeleton
of the regular coclass tree
for
(a special case of a pruned coclass tree, see §7) and omitted the irregular component
whose members are entirely non-metabelian of derived length 3.
4. Definitions and terminology
According to Newman ([20] , 2, pp. 52-53), there exist several distinct definitions of the parent
of a finite
-group
. The common principle is to form the quotient
of
by a suitable normal subgroup
which can be either
[(P)]
1) the centre
of
, whence
is called central quotient of
or
2) the last non-trivial term
of the lower central series of
, where
denotes the nilpotency class of
or
3) the last non-trivial term
of the lower exponent-
central series of
, where
denotes the exponent-p class of
or
4) the last non-trivial term
of the derived series of
, where
denotes the derived length of
.
In each case,
is called an immediate descendant of
and a directed edge of the tree is defined either by
in the direction of the canonical projection
onto the quotient
or by
in the opposite direction, which is more usual for descendant trees. The former convention is adopted by Leedham-Green and Newman ([21] , 2, pp. 194-195), by du Sautoy and Segal ([22] , 7, p. 280), by Leedham-Green and McKay ([23] , Dfn.8.4.1, p. 166), and by Eick, Leedham-Green, Newman and O’Brien ([24] , 1). The latter definition is used by Newman ([20] , 2, pp. 52-53), by Newman and O’Brien ([25] , 1, p. 131), by du Sautoy ([1] , 1, p. 67), by Dietrich, Eick and Feichtenschlager ([26] , 2, p. 46) and by Eick and Leedham-Green ([2] , 1, p. 275).
In the following, the direction of the canonical projections is selected for all edges. Then, more generally, a vertex
is a descendant of a vertex
, and
is an ancestor of
, if either
is equal to
or there is a path
(1)
of directed edges from
to
. The vertices forming the path necessarily coincide with the iterated parents
of
, with
:
(2)
In the most important special case (P2) of parents defined as last non-trivial lower central quotients, they can also be viewed as the successive quotients
of class
of
when the nilpotency class of
is given by
:
(3)
with
.
Generally, the descendant tree
of a vertex
is the subtree of all descendants of
, starting at the root
. The maximal possible descendant tree
of the trivial group 1 contains all finite
-groups and is somewhat exceptional, since, for any parent definition (P1 - P4), the trivial group 1 has infinitely many abelian
-groups as its immediate descendants. The parent definitions (P2 - P3) have the advantage that any non-trivial finite
-group (of order divisible by
) possesses only finitely many immediate descendants.
5. Pro-pgroups and coclass trees
For a sound understanding of coclass trees as a particular instance of descendant trees, it is necessary to sum- marize some facts concerning infinite topological pro-
groups. The members
, with
, of the lower central series of a pro-
group
are open and closed subgroups of finite index, and therefore the corresponding quotients
are finite
-groups. The pro-
group
is said to be of coclass
when the limit
of the coclass of the successive quotients exists and is
finite. An infinite pro-
group
of coclass
is a
-adic pre-space group ([23] , Dfn.7.4.11, p. 147), since it has a normal subgroup
, the translation group, which is a free module over the ring
of
-adic integers of uniquely determined rank
, the dimension, such that the quotient
is a finite
-group, the point group, which acts on
uniserially. The dimension is given by
(4)
A central finiteness result for infinite pro-
groups of coclass
is provided by the so-called Theorem D, which is one of the five Coclass Theorems proved in 1994 independently by Shalev [27] and by Leedham-Green ([28] , Thm.7.7, p. 66), and conjectured in 1980 already by Leedham-Green and Newman ([21] , 2, pp. 194- 196). Theorem D asserts that there are only finitely many isomorphism classes of infinite pro-
groups of coclass
, for any fixed prime
and any fixed non-negative integer
. As a consequence, if
is an infinite pro-
group of coclass
, then there exists a minimal integer
such that the following three conditions are satisfied for any integer
.
・
;
・
is not a lower central quotient of any infinite pro-
group of coclass
which is not isomorphic
to
;
・
is cyclic of order
.
The descendant tree
, with respect to the parent definition (P2), of the root
with
minimal
is called the coclass tree
of
and its unique maximal infinite (reverse-directed) path
(5)
is called the mainline (or trunk) of the tree.
6. Tree Diagram
Further terminology, used in diagrams visualizing finite parts of descendant trees, is explained in Figure 1 by means of an artificial abstract tree. On the left hand side, a level indicates the basic top-down design of a descendant tree. For concrete trees, such as those in Figure 2, resp. Figure 3, etc., the level is usually replaced by a scale of orders increasing from the top to the bottom. A vertex is capable (or extendable) if it has at least one immediate descendant, otherwise it is terminal (or a leaf). Vertices sharing a common parent are called siblings.
![]()
Figure 1. Terminology for descendant trees.
If the descendant tree is a coclass tree
with root
and with mainline vertices ![]()
labelled according to the level n, then the finite subtree defined as the difference set
(6)
is called the nth branch (or twig) of the tree or also the branch
with root
, for any
. The depth of a branch is the maximal length of the paths connecting its vertices with its root.
Figure 1 shows a descendant tree whose branches
both have depth 0, and
, resp.
, are isomorphic as trees.
If all vertices of depth bigger than a given integer
are removed from branch
, then we obtain the
(depth-)pruned branch
. Correspondingly, the pruned coclass tree
, resp. the entire coclass tree
, consists of the infinite sequence of its pruned branches
, resp. branches
,
connected by the mainline, whose vertices
are called infinitely capable.
7. Virtual Periodicity
The periodicity of branches of depth-pruned coclass trees has been proved with analytic methods using zeta functions ([22] , 7, Thm.15, p. 280) of groups by du Sautoy ([1] , Thm.1.11, p. 68, and Thm.8.3, p. 103), and with algebraic techniques using cohomology groups by Eick and Leedham-Green [2] . The former methods admit the qualitative insight of ultimate virtual periodicity, the latter techniques determine the quantitative structure.
Theorem 7.1 For any infinite pro-
group
of coclass
and dimension
, and for any given depth
, there exists an effective minimal lower bound
, where periodicity of length
of depth-
pruned branches of the coclass tree
sets in, that is, there exist graph isomorphisms
(7)
Proof. The graph isomorphisms of depth-
pruned banches with roots of sufficiently large order
are derived with cohomological methods in ([2] , Thm.6, p. 277, Thm.9, p. 278) and the effective lower bound
for the branch root orders is established in ([2] , Thm.29, p. 287).
This central result can be expressed ostensively: When we look at a coclass tree through a pair of blinkers and ignore a finite number of pre-periodic branches at the top, then we shall see a repeating finite pattern (ultimate periodicity). However, if we take wider blinkers the pre-periodic initial section may become longer (virtual periodicity).
The vertex
is called the periodic root of the pruned coclass tree, for a fixed value of the depth
.
See Figure 1.
8. Multifurcation and coclass Graphs
Assume that parents of finite
-groups are defined as last non-trivial lower central quotients (P2). For a
-group G of coclass
, we can distinguish its (entire) descendant tree
and its coclass-r descendant tree
, the subtree consisting of descendants of coclass r only. The group G is coclass-settled if
.
The nuclear rank
of G (see §14) in the theory of the
-group generation algorithm by Newman [8] and O’Brien [9] provides the following criteria.
・
is terminal, and thus trivially coclass-settled, if and only if
;
・ If
, then G is capable, but it remains unknown whether G is coclass-settled;
・ If
, then G is capable and certainly not coclass-settled.
In the last case, a more precise assertion is possible: If G has coclass
and nuclear rank
, then it gives rise to an
-fold multifurcation into a regular coclass-
descendant tree
and
irregular descendant graphs
of coclass
, for
. Consequently, the descendant tree of G is the disjoint union
(8)
Multifurcation is correlated with different orders of the last non-trivial lower central of immediate descendants. Since the nilpotency class increases exactly by a unit,
, from a parent
to any immediate descendant
, the coclass remains stable,
, if
.
In this case,
is a regular immediate descendant with directed edge
of depth 1, as usual. However,
the coclass increases by
, if
with
. Then
is called an irregular immediate
descendant with directed edge of depth
.
If the condition of depth (or step size) 1 is imposed on all directed edges, then the maximal descendant tree
of the trivial group 1 splits into a countably infinite disjoint union
(9)
of directed coclass graphs
, which are rather forests than trees. More precisely, the above mentioned
Coclass Theorems imply that
(10)
is the disjoint union of finitely many coclass trees
of pairwise non-isomorphic infinite pro-
groups
of coclass
(Theorem D) and a finite subgraph
of sporadic groups lying outside of any coclass tree.
9. Identifiers
The SmallGroups Library identifiers of finite groups, in particular p-groups, given in the form
![]()
in the following concrete examples of descendant trees, are due to Besche, Eick and O’Brien [16] [17] . When the group orders are given in a scale on the left hand side as in Figure 2 and Figure 3, the identifiers are briefly denoted by
![]()
Depending on the prime p, there is an upper bound on the order of groups for which a SmallGroup identifier exists, e.g.
for
, and
for
. For groups of bigger orders, a notation with generalized identifiers resembling the descendant structure is employed: a regular immediate descendant, con- nected by an edge of depth 1 with its parent P, is denoted by
![]()
and an irregular immediate descendant, connected by an edge of depth
with its parent P, is denoted by
![]()
The ANUPQ package [11] containing the implementation of the p-group generation algorithm uses this notation, which goes back to Ascione in 1979 [18] .
10. Concrete Examples of Trees
In all examples, the underlying parent definition (P2) corresponds to the usual lower central series. Occasional differences to the parent definition (P3) with respect to the lower exponent-p central series are pointed out.
10.1. Coclass 0
The coclass graph
(11)
of finite p-groups of coclass 0 does not contain a coclass tree and consists of the trivial group 1 and the cyclic group
of order p, which is a leaf (however, it is capable with respect to the lower exponent-p central series). For
the SmallGroup identifier of
is
, for
it is
.
10.2. Coclass 1
The coclass graph
(12)
of finite p-groups of coclass 1 consists of the unique coclass tree with root
, the elementary
abelian p-group of rank 2, and a single isolated vertex (a terminal orphan without proper parent in the same co- class graph, since the directed edge to the trivial group 1 has depth 2), the cyclic group
of order
in the sporadic part
(however, this group is capable with respect to the lower exponent-
central series). The tree
is the coclass tree of the unique infinite pro-p group
of coclass 1.
For
, resp.
, the SmallGroup identifier of the root
is
, resp.
, and a tree diagram of the coclass graph from branch
up to branch
(counted with respect to the p-logarithm of the order of the branch root) is drawn in Figure 2, resp. Figure 3, where all groups of order at least
are metabelian, that is non-abelian with derived length 2 (vertices represented by black discs in contrast to contour squares indicating abelian groups). In Figure 3, smaller black discs denote metabelian 3-groups where even the maximal subgroups are non-abelian, a feature which does not occur for the metabelian 2-groups in Figure 2, since they all possess an abelian subgroup of index p (usually exactly one). The coclass tree of
, resp.
, has periodic root
and period of length 1 starting with branch
, resp. periodic root
and period of length 2 starting with branch
. Both trees have branches of bounded depth 1, so their virtual periodicity is in fact a strict periodicity. The
, resp.
, denote isoclinism families [29] [30] .
However, the coclass tree of
with
has unbounded depth and contains non-metabelian groups, and the coclass tree of
with
has even unbounded width, that is the number of descendants of a fixed order increases indefinitely with growing order [26] .
With the aid of kernels and targets of Artin transfer homomorphisms [3] , the diagrams in Figure 2 and Figure 3 can be endowed with additional information and redrawn as structured descendant trees ([6] , Figure 3.1, p. 419, and Figure 3.2, p. 422).
The concrete examples
and
provide an opportunity to give a parametrized polycyclic power-commutator presentation ([31] , pp. 82-84) for the complete coclass tree, mentioned in §2 as a benefit of the descendant tree concept and as a consequence of the periodicity of the pruned coclass tree. In both cases, the group
is generated by two elements
but the presentation contains the series of higher commutators
,
, starting with the main commutator
. The nilpotency is formally expressed
by
, when the group is of order
.
For
, there are two parameters
and the pc-presentation is given by
(13)
The 2-groups of maximal class, that is of coclass 1, form three periodic infinite sequences:
・ the dihedral groups,
,
, forming the mainline (with infinitely capable vertices);
・ the generalized quaternion groups,
,
, which are all terminal vertices;
・ the semidihedral groups,
,
, which are also leaves.
For
, there are three parameters
and
and the pc-presentation is given by
(14)
-groups with parameter
possess an abelian maximal subgroup, those with parameter
do not. More precisely, an existing abelian maximal subgroup is unique, except for the two extra special groups
and
, where all four maximal subgroups are abelian.
In contrast to any bigger coclass
, the coclass graph
exclusively contains
-groups
with abelianization
of type
, except for its unique isolated vertex. The case
is distinguished by the truth of the reverse statement: Any
-group with abelianization of type (2,2) is of coclass 1 (Taussky’s Theorem ([32] , p. 83).
Figure 4 shows the interface between finite 3-groups of coclass 1 and 2 of type (3,3).
10.3. Coclass 2
The genesis of the coclass graph
with
is not uniform.
-groups with several distinct abelia- nizations contribute to its constitution. For coclass
, there are essential contributions from groups
with abelianizations
of the types
,
,
, and an isolated contribution by the cyclic group of order
:
(15)
10.3.1. Abelianization of type ![]()
As opposed to
-groups of coclass
with abelianization of type
or
, which arise as regular descendants of abelian
-groups of the same types,
-groups of coclass
with abelianization of type
arise from irregular descendants of a non-abelian
-group with coclass
and nuclear rank 2.
For the prime
, such groups do not exist at all, since the dihedral group
is coclass-settled, which is the deeper reason for Taussky’s Theorem. This remarkable fact has been observed by Bagnera ([33] , Part 2, 4, p. 182) in 1898 already.
For odd primes
, the existence of
-groups of coclass 2 with abelianization of type
is due to the fact that the extra special group
is not coclass-settled. Its nuclear rank equals 2, which gives rise to a bifurcation of the descendant tree
into two coclass graphs. The regular component
is a subtree of the unique tree
in the coclass graph
. The irregular
component
becomes a subgraph
of the coclass graph
when the
connecting edges of depth 2 of the irregular immediate descendants of
are removed.
For
, this subgraph
is drawn in Figure 4. It has seven top level vertices of three important kinds, all having order
, which have been discovered by Bagnera ([33] , Part 2, 4, pp. 182-183).
![]()
Figure 4. 3-groups of coclass 2 with abelianization (3,3).
・ Firstly, there are two terminal Schur
-groups [34]
and
in the sporadic part
of the coclass graph
;
・ Secondly, the two groups
and
are roots of finite trees
in the sporadic part
(however, since they are not coclass-settled, the complete trees
are infinite);
・ And, finally, the three groups
,
and
give rise to (infinite) coclass trees, e.g.,
,
,
, each having a metabelian mainline, in the coclass graph
. None of these three groups is coclass-settled. See §21.
Displaying additional information on kernels and targets of Artin transfers [3] , we can draw these trees as structured descendant trees ([6] , Figure 3.5, p. 439, Figure 3.6, p. 442, and Figure 3.7, p. 443).
Definition 10.1 Generally, a Schur group (called a closed group by I. Schur, who coined the concept) is a
pro-p group G whose relation rank
coincides with its generator rank
. A
-group is a pro-p group G which possesses an automorphism ![]()
inducing the inversion
on its abelianization
. A Schur
-group [7] [34] -[36] is a Schur group G which is also a
-group and has a finite abelianization
.
It should be pointed out that
is not root of a coclass tree, since its immediate descendant
, which is root of a coclass tree with metabelian mainline vertices, has two siblings
, resp.
, which give rise to a single, resp. three, coclass tree(s) with non-metabelian mainline vertices having cyclic centres of order 3 and branches of considerable complexity but nevertheless of bounded depth 5.
10.3.2. Pro-3 groups of Coclass 2 with Non-trivial centre
Eick, Leedham-Green, Newman and O’Brien ([24] , 4, Thm.4.1) have constructed a family of infinite pro-3 groups with coclass 2 having a non-trivial centre of order 3. The members are characterized by three parameters
. Their finite quotients generate all mainline vertices with bicyclic centres of type
of six coclass trees in the coclass graph
. The association of parameters to the roots of these six trees is given in Table 1, the tree diagrams are indicated in Figure 4 and Figure 5, and the parametrized pro-3 presentation is given by
(16)
Figure 5 shows some finite 3-groups with coclass 2 and type
.
10.3.3. Abelianization of type ![]()
For
, the top levels of the subtree
of the coclass graph
are drawn in Figure 5. The most important vertices of this tree are the eight siblings sharing the common parent
, which are of three important kinds.
・ Firstly, there are three leaves
,
,
having cyclic centre of order 9, and a single leaf
with bicyclic centre of type
;
・ Secondly, the group
is root of a finite tree
;
・ And, finally, the three groups
,
and
give rise to infinite coclass trees, e. g.,
,
,
, each having a metabelian mainline, the first with cyclic centres of order
, the second and third with bicyclic centres of type
.
Here, it should be emphasized that
is not root of a coclass tree, since aside from its descendant
, which is root of a coclass tree with metabelian mainline vertices, it possesses five further descen- dants which give rise to coclass trees with non-metabelian mainline vertices having cyclic centres of order 3 and branches of considerable complexity, here partially even with unbounded depth ([24] , Thm.4.2(a-b)).
10.3.4. Abelianization of type ![]()
For
, resp.
, there exists a unique coclass tree with
-groups of type
in the coclass graph
. Its root is the elementary abelian
-group of type
, that is,
, resp.
.
![]()
Figure 5. 3-groups of coclass 2 with abelianization (9,3).
This unique tree corresponds to the pro-2 group of the family #59 by Newman and O’Brien ([25] , Appendix A, no. 59, p. 153, Appendix B, Tbl. 59, p. 165), resp. the pro-3 group given by the parameters
in Table 1. For
, the tree is indicated in Figure 6.
Figure 6 shows some finite 2-groups of coclass 2,3,4 and type (2,2,2).
10.4. Coclass 3
Here again,
-groups with several distinct abelianizations contribute to the constitution of the coclass graph
. There are regular, resp. irregular, essential contributions from groups
with abelianizations
of the types
,
,
,
, resp.
,
,
, and an isolated contribution by the cyclic group of order
.
10.4.1. Abelianization of type ![]()
Since the elementary abelian
-group
of rank 3, that is,
, resp.
, for
, resp.
![]()
Figure 6. 2-groups of coclass 3 with abelianization (2,2,2).
, is not coclass-settled, it gives rise to a multifurcation. The regular component
has
been described in the section about coclass 2. The irregular component
becomes a
subgraph
of the coclass graph
when the connecting edges of depth 2 of the
irregular immediate descendants of
are removed.
For
, this subgraph
is contained in Figure 6. It has nine top level vertices of order
which can be divided into terminal and capable vertices:
・ the groups
and
are leaves;
・ the five groups
and the two groups
are infinitely capable.
The trees arising from the capable vertices are associated with infinite pro-2 groups by Newman and O’Brien ([25] , Appendix A, no. 73-79, pp. 154-155, and Appendix B, Tbl. 73-79, pp. 167-168) in the following manner:
gives rise to
associated with family
, and
associated with family
;
is associated with family
;
is associated with family
;
is associated with family
;
gives rise to
associated with family
(see §21), and finally
is associated with family
(see Figure 6).
The roots of the coclass trees
in Figure 6 and
in Figure 7 are
siblings.
10.4.2. Hall-Senior classification
Seven of these nine top level vertices have been investigated by Benjamin, Lemmermeyer and Snyder ([37] , 2, Tbl. 1) with respect to their occurrence as class-2 quotients
of bigger metabelian 2-groups
of type
and with coclass 3, which are exactly the members of the descendant trees of the seven vertices. These authors use the classification of 2-groups by Hall and Senior [29] which is put in correspondence with the SmallGroups Library [16] [17] in Table 2. The complexity of the descendant trees of these seven vertices increases with the 2-ranks and 4-ranks indicated in Table 2, where the maximal subgroups of index 2 in
are denoted by
, for
.
11. History of Descendant Trees
Descendant trees with central quotients as parents (P1) are implicit in Hall’s 1940 paper [30] about isoclinism of groups. Trees with last non-trivial lower central quotients as parents (P2) were first presented by Leedham- Green at the International Congress of Mathematicians in Vancouver, 1974 [20] . The first extensive tree diagrams have been drawn manually by Ascione, Havas and Leedham-Green (1977) [14] , by Ascione (1979) [18] and by Nebelung (1989) [15] . In the former two cases, the parent definition by means of the lower exponent-p central series (P3) was adopted in view of computational advantages, in the latter case, where theoretical aspects were focused, the parents were taken with respect to the usual lower central series (P2).
The kernels and targets of Artin transfer homomorphisms have recently turned out to be compatible with parent-descendant relations between finite p-groups and can favourably be used to endow descendant trees with additional structure [6] .
12. The Construction: p-group Generation algorithm
The p-group generation algorithm by Newman [8] and O’Brien [9] [10] is a recursive process for constructing the descendant tree of an assigned finite p-group which is taken as the root of the tree. It is discussed in some detail in §§13-19.
13. Lower Exponent-p Central Series
For a finite p-group G, the lower exponent-p central series (briefly lower p-central series) of
is a
descending series
of characteristic subgroups of
, defined recursively by
(17)
Since any non-trivial finite p-group
is nilpotent, there exists an integer
such that
and
is called the exponent-p class (briefly p-class) of
. Only the trivial
![]()
Table 2. Class-2 quotients
of certain metabelian 2-groups
of type (2,2,2).
group 1 has
. Generally, for any finite p-group
, its p-class can be defined as
.
The complete lower p-central series of
is therefore given by
(18)
since
is the Frattini subgroup of
.
For the convenience of the reader and for pointing out the shifted numeration, we recall that the (usual) lower
central series of
is also a descending series
of characteristic subgroups of
, defined
recursively by
(19)
As above, for any non-trivial finite
-group
, there exists an integer
such that
and
is called the nilpotency class of
, whereas
is called the index of
nilpotency of
. Only the trivial group 1 has
.
Thus, the complete lower central series of
is given by
(20)
since
is the commutator subgroup or derived subgroup of
.
The following rules should be remembered for the exponent-
class:
Let
be a finite
-group.
[(R)]
1)
, since the
descend more quickly than the
;
2)
, for some group
, for any
;
3) For any
, the conditions
and
imply
;
4) For any
,
, for all
, in particular,
, for all
.
We point out that every non-trivial finite
-group
defines a maximal path with respect to the parent definition (P3), consisting of
edges,
(21)
and ending in the trivial group
. The last but one quotient of the maximal path of
is the
elementary abelian
-group
of rank
, where
denotes the generator rank of
.
14. p-covering group, p-multiplicator and nucleus
Let
be a finite
-group with
generators. Our goal is to compile a complete list of pairwise non- isomorphic immediate descendants of
. It turned out that all immediate descendants can be obtained as quotients of a certain extension
of
which is called the
-covering group of
and can be constructed in the following manner.
We can certainly find a presentation of
in the form of an exact sequence
(22)
where
denotes the free group with
generators and
is an epimorphism with kernel
. Then
is a normal subgroup of
consisting of the defining relations for
. For
elements
and
, the conjugate
and thus also the commutator ![]()
are contained in
. Consequently,
is a characteristic subgroup of
, and the
-multiplicator
of
is an elementary abelian
-group, since
(23)
Now we can define the
-covering group of
by
(24)
and the exact sequence
(25)
shows that
is an extension of
by the elementary abelian
-multiplicator. We call
(26)
the
-multiplicator rank of
.
Let us assume now that the assigned finite
-group
is of
-class
. Then the
conditions
and
imply
, according to the rule (R3), and we can define the
nucleus of
by
(27)
as a subgroup of the
-multiplicator. Consequently, the nuclear rank
(28)
of
is bounded from above by the
-multiplicator rank.
15. Allowable subgroups of the p-multiplicator
As before, let
be a finite p-group with
generators. Any
-elementary abelian central extension
of G by a p-elementary abelian subgroup
such that ![]()
is a quotient of the
-covering group
of
.
The reason is that there exists an epimorphism
such that
, where ![]()
denotes the canonical projection. Consequently, we have
and thus
. Further,
, since
is
-elementary, and
,
since
is central. Together this shows that
and thus
induces the desired
epimorphism
such that
.
In particular, an immediate descendant
of
is a
-elementary abelian central extension
(29)
of
, since
![]()
where
.
A subgroup
of the
-multiplicator of
is called allowable if it is given by the kernel
of an epimorphism
onto an immediate descendant
of
. An equivalent
characterization is that
is a proper subgroup which supplements the nucleus
(30)
Therefore, the first part of our goal to compile a list of all immediate descendants of
is done, when we
have constructed all allowable subgroups of
which supplement the nucleus
,
where
. However, in general the list
(31)
where
will be redundant, due to isomorphisms ![]()
among the immediate descendants.
16. Orbits under extended Automorphisms
Two allowable subgroups
and
are called equivalent if the quotients
, that
are the corresponding immediate descendants of
, are isomorphic.
Such an isomorphism
between immediate descendants of
with ![]()
has the property that
![]()
and thus induces an automorphism
of
which can be extended to an automorphism
of the
-covering group
of
. The restriction of this extended automorphism
to the
-multiplicator
of
is determined uniquely by
. Since
![]()
according to the rule (R2), each extended automorphism
induces a permutation
of the allowable subgroups
. We define
(32)
to be the permutation group generated by all permutations induced by automorphisms of
. Then the map
,
is an epimorphism and the equivalence classes of allowable subgroups ![]()
are precisely the orbits of allowable subgroups under the action of the permutation group
.
Eventually, our goal to compile a list
of all immediate descendants of
will be done,
when we select a representative
for each of the
orbits of allowable subgroups of
under the action of
. This is precisely what the
-group generation algorithm does in a single step of the recursive procedure for constructing the descendant tree of an assigned root.
17. Capable p-groups and step Sizes
We recall from §6 that a finite p-group G is called capable (or extendable) if it possesses at least one immediate descendant, otherwise it is called terminal (or a leaf). As mentioned in §8 already, the nuclear rank
of G admits a decision about the capability of G:
・ G is terminal if and only if
;
・ G is capable if and only if
.
In the case of capability,
has immediate descendants of
different step sizes
, in dependence on the index
(33)
of the corresponding allowable subgroup
in the
-multiplicator
. When G is of order
, then an immediate descendant of step size
is of order
![]()
For the related phenomenon of multifurcation of a descendant tree at a vertex G with nuclear rank
see §8 on multifurcation and coclass graphs.
The
-group generation algorithm provides the flexibility to restrict the construction of immediate descen- dants to those of a single fixed step size
, which is very convenient in the case of huge descendant numbers (see the next section).
18. Numbers of immediate Descendants
We denote the number of all immediate descendants, resp. immediate descendants of step size
, of G by
, resp.
. Then we have
. As concrete examples, we present some interesting finite
metabelian
-groups with extensive sets of immediate descendants, using the SmallGroups identifiers and additionally pointing out the numbers
of capable immediate descendants in the usual format
(34)
as given by actual implementations of the
-group generation algorithm in the computational algebra systems GAP and MAGMA. These invariants completely determine the local structure of the descendant tree
.
First, let
. We begin with groups having abelianization of type
. See Figure 6.
・ The group
of coclass 3 has ranks
,
and descendant numbers
,
. See
21;
・ The group
of coclass 3 has ranks
,
and descendant numbers
,
. See §21;
・ The group
of coclass 3 has ranks
,
and
descendant numbers
,
;
Next, let
. We consider groups having abelianization of type
. See Figure 4;
・ The group
of coclass 1 has ranks
,
and descendant numbers
,
;
・ The group
of coclass 2 has ranks
,
and descendant numbers
,
;
・ One of its immediate descendants, the group
, has ranks
,
and descendant numbers
,
.
In contrast, groups with abelianization of type
are partially located beyond the limit of actual computability.
・ The group
of coclass 2 has ranks
,
and descendant numbers
,
;
・ The group
of coclass 3 has ranks
,
and descendant numbers
,
unknown;
・ The group
of coclass 4 has ranks
,
and descendant numbers
,
unknown.
19. Schur multiplier
Via the isomorphism
![]()
group
(35)
can be viewed as the additive analogue of the multiplicative group
(36)
of all roots of unity.
Let p be a prime number and G be a finite p-group with presentation
as in the previous section. Then the second cohomology group
(37)
of the G-module
is called the Schur multiplier of G. It can also be interpreted as the quotient group
(38)
Shafarevich ([38] , 6, p. 146) has proved that the difference between the relation rank
of G and the generator rank
of G is given by the
minimal number of generators of the Schur multiplier of G, that is
(39)
Boston and Nover ([39] , 3.2, Prop. 2) have shown that
(40)
for all quotients
of p-class
,
, of a pro-p group G with finite abelianization
.
Furthermore, Blackhurst (in the appendix On the nucleus of certain p-groups of a paper by Boston, Bush and Hajir [35] ) has proved that a non-cyclic finite p-group G with trivial Schur multiplier
is a terminal vertex in the descendant tree
of the trivial group 1, that is,
(41)
We conclude this section by giving two examples.
・ A finite p-group G has a balanced presentation
if and only if
, that is, if and only if its Schur multiplier
is trivial. Such a group
is called a Schur group [7] [34] -[36] and it must be a leaf in the descendant tree
;
・ A finite p-group G satisfies
if and only if
, that is, if
and only if it has a non-trivial cyclic Schur multiplier
. Such a group is called a Schur
group.
20. Pruning strategies
For searching a particular group in a descendant tree
by looking for patterns defined by the kernels and targets of Artin transfer homomorphisms [6] , it is frequently adequate to reduce the number of vertices in the branches of a dense tree with high complexity by sifting groups with desired special properties, for example
[(F)]
1) filtering the
-groups (see Definition 10.1);
2) eliminating a set of certain transfer kernel types (TKTs, see ([6] , pp. 403-404));
3) cancelling all non-metabelian groups (thus restricting to the metabelian skeleton);
4) removing metabelian groups with cyclic centre (usually of higher complexity);
5) cutting off vertices whose distance from the mainline (depth) exceeds some lower bound;
6) combining several different sifting criteria.
The result of such a sieving procedure is called a pruned descendant tree
with respect to the desired set of properties.
However, in any case, it should be avoided that the mainline of a coclass tree is eliminated, since the result would be a disconnected infinite set of finite graphs instead of a tree. We expand this idea further in the following detailed discussion of new phenomena.
21. Striking News: periodic bifurcations intrees
We begin this section about brand-new discoveries with the most recent example of periodic bifurcations in trees of 2-groups. It has been found on the 17th of January 2015, motivated by a search for metabelian 2-class tower groups [40] of complex quadratic fields [41] and complex bicyclic biquadratic Dirichlet fields [42] .
21.1. Finite 2-Groups G with ![]()
The 2-groups under investigation are three-generator groups with elementary abelian commutator factor group of type
. As shown in Figure 6 of §10, all such groups are descendants of the abelian root
. Among its immediate descendants of step size 2, there are three groups which reveal multifurcation.
has nuclear rank
, giving rise to 3-fold multifurcation. The two groups
and
possess the required nuclear rank
for bifurcation. Due to the arithmetical origin of the problem, we focused on the latter,
, and constructed an extensive finite part of its pruned descendant tree
, using the
-group generation algorithm [8] -[10] as implemented in the computational algebra system Magma [13] [43] [44] . All groups turned out to be metabelian.
Remark 21.1 Since our primary intention is to provide a sound group theoretic background for several phe- nomena discovered in class field theory and algebraic number theory, we eliminated superfluous brushwood in the descendant trees to avoid unnecessary complexity.
The selected sifting process for reducing the entire descendant tree
to the pruned descendant tree
filters all vertices which satisfy one of the conditions in Equation (44) or (49), and essentially consists of pruning strategy (F2), more precisely, of
1) omitting all the 14 terminal step size-2 descendants, and 5, resp. 4, of the 6 capable step size-2 descendants, together with their complete descendant trees, in Theorem 21.1, resp. Corollary 21.1, and
2) eliminating all, resp. 4, of the 5 terminal step size-1 descendants in Theorem 21.1, resp. Corollary 21.1.
Denote by
the generators of a finite 2-group
with abelian type invariants
. We fix an ordering of the seven maximal normal subgroups by putting
(42)
Just within this subsection, we select a special designation for a TKT [[6] , p. 403-404] whose first layer consists exactly of all these seven planes in the 3-dimensional
-vector space
, in any ordering.
Definition 21.1 The transfer kernel type (TKT)
is called a permutation if all seven
members of the first layer
are maximal subgroups of
and there exists a permutation
such that
.
For brevity, we give 2-logarithms of abelian type invariants in the following theorem and we denote iteration
by formal exponents, for instance,
,
,
and
. Further, we eliminate an initial anomaly of generalized identifiers by putting
and
, formally.
Theorem 21.1 Let
be a positive integer bounded from above by 10.
1) In the descendant tree
of
, there exists a unique path of length
,
![]()
of (reverse) directed edges with uniform step size 2 such that
, for all ![]()
(along the path,
is a section of the surjection
), and all the vertices
(43)
of this path share the following common invariants:
・ the transfer kernel type with layer
containing three 2-cycles (and nearly a permutation, except for the
first component which is total,
),
(44)
・ the 2-multiplicator rank and the nuclear rank, giving rise to the bifurcation,
(45)
・ and the counters of immediate descendants,
(46)
determining the local structure of the descendant tree.
2) A few other invariants of the vertices
depend on the superscript
,
・ the 2-logarithm of the order, the nilpotency class and the coclass,
(47)
・ a single component of layer
, three components of layer
, and layer
of the transfer target type
(48)
In view of forthcoming number theoretic applications, we add the following
Corollary 21.1 Let
be a non-negative integer.
1) The regular component
of the descendant tree
is a coclass tree which
contains a unique periodic sequence whose vertices
with
are characte-
rized by a permutation TKT
(49)
with a single fixed point
and the same three 2-cycles
,
,
as in the mainline TKT of Equation (44).
2) The irregular component
of the descendant tree
is a forest which contains a
unique second coclass tree
whose mainline vertices ![]()
with
possess the same permutation TKT as in Equation (49), apart from the first coclass tree
, where
, whose mainline vertices
with ![]()
share the TKT in Equation (44).
Proof. (of Theorem 21.1, Corollary 21.1 and Theorem 21.2)
The
-group generation algorithm [8] -[10] as implemented in the Magma computational algebra system [13] [43] [44] was employed to construct the pruned descendant tree
with root
which we
defined as the disjoint union of all pruned coclass trees
with the successive descendants
,
, of step size 2 of
as roots. Using the well-known virtual periodicity [1]
[2] of each coclass tree
, which turned out to be strict and of the smallest possible length 1, the
vertical construction was terminated at nilpotency class 12, considerably deeper than the point where periodicity sets in. The horizontal construction was extended up to coclass 13, where the amount of CPU time started to become annoying.
Within the frame of our computations, the periodicity was not restriced to bifurcations only: It seems that the pruned (or maybe even the entire) descendant trees
are all isomorphic to
as graphs. This is visualized impressively by Figure 7.
The extent to which we constructed the pruned descendant tree suggests the following conjecture.
Conjecture 21.1 Theorem 21.1, Corollary 21.1 and Theorem 21.2 remain true for an arbitrarily large positive integer
, not necessarily bounded by 10.
Remark 21.2 We must emphasize that the root
in Figure 7 is drawn for the sake of completeness
only, and that the mainline of the coclass tree
is exceptional, since
・ its root is not a descendant of
and
・ the TKT of its vertices
with
,
(50)
is a permutation with 5 fixed points and only a single 2-cycle
.
One-parameter polycyclic pc-presentations for all occurring groups are given as follows.
1) For the mainline vertices of the coclass tree
with class
, that is, starting with
and excluding the root
, by
(51)
2) For the mainline vertices of the coclass tree
with class
by
(52)
3) For the mainline vertices of the coclass tree
with class
by
(53)
Theorem 21.2 For higher coclass
the presentations (52) and (53) can be generalized in the shape of a two-parameter polycyclic pc-presentation for class
.
(54)
To obtain a presentation for the vertices
,
, at depth 1 in the distinguished
periodic sequence whose vertices are characterized by the permutation TKT (49), we must only add the single
relation
to the presentation (54) of the mainline vertices of the coclass tree
given in
Theorem 21.2.
21.2. Finite 3-groups G with ![]()
We continue this section with periodic bifurcations in trees of 3-groups, which have been discovered in 2012 and 2013 [45] -[47] , inspired by a search for 3-class tower groups of complex quadratic fields [7] [48] [49] , which must be Schur
-groups.
These 3-groups are two-generator groups of coclass at least 2 with elementary abelian commutator quotient of type
. As shown in Figure 4 of §10, all such groups are descendants of the extra special group
. Among its 7 immediate descendants of step size 2, there are only two groups which satisfy the requirements arising from the arithmetical background.
The two groups
and
do not show multifurcation themselves but they are not coclass- settled either, since their immediate mainline descendants
and
possess the required nuclear rank
for bifurcation. We constructed an extensive finite part of their pruned descendant trees
,
, using the p-group generation algorithm [8] -[10] as implemented in the computational algebra system Magma [13] [43] [44] .
Denote by
the generators of a finite 3-group
with abelian type invariants
. We fix an ordering of the four maximal normal subgroups by putting
(55)
Within this subsection, we make use of special designations for transfer kernel types (TKTs) which were defined generally in [[6] , p. 403-404] and more specifically for the present scenario in [4] [50] .
We are interested in the unavoidable mainline vertices with TKTs c.18,
, resp. c.21,
, and, above all, in most essential vertices of depth 1 forming periodic sequences with TKTs
,
and
,
, resp.
,
and
,
, and we want to eliminate the numerous and annoying vertices with TKTs
,
, resp.,.
We point out that, for instance
,
, is a shortcut for the layer
of the complete (layered) TKT
.
Remark 21.3 We choose the following sifting strategy for reducing the entire descendant tree
to the pruned descendant tree
. We filter all vertices which, firstly, are
-groups, and secondly satisfy one of the conditions in Equations (58) or (67), whence the process is a combination (F6) = (F1) + (F2) + (F5) and consists of
1) keeping all of the 3 terminal step size-2 descendants, which are exactly the Schur
-groups, and omitting 2 of the 3 capable step size-2 descendants having TKT H.4, resp.
, together with their complete descendant trees, and
2) eliminating 2 of the 5 terminal step size-1 descendants having TKT
, resp.
, and 2 of the 3 capable step size-1 descendants having TKT
, resp.
, in Theorem 21.3.
For brevity, we give 3-logarithms of abelian type invariants in the following theorem and we denote iteration
by formal exponents, for instance,
,
,
, and
. Further, we eliminate some initial anomalies of generalized identifiers by putting
, ,
, ,
,
, formally.
Theorem 21.3 Let
be a positive integer bounded from above by 8.
1) In the descendant tree
of
, resp.
, there exists a unique path of length
,
![]()
of (reverse) directed edges of alternating step sizes 1 and 2 such that
, for all
, and all the vertices with even superscript
,
,
(56)
resp. all the vertices with odd superscript
,
(57)
of this path share the following common invariants, respectively:
・ the uniform (w.r.t. i) transfer kernel type, containing a total component
,
(58)
・ the 2-multiplicator rank and the nuclear rank,
(59)
resp., giving rise to the bifurcation for odd
,
(60)
・ and the counters of immediate descendants,
(61)
resp.
(62)
determining the local structure of the descendant tree.
2) A few other invariants of the vertices
depend on the superscript i,
・ the 3-logarithm of the order, the nilpotency class and the coclass,
(63)
resp.
(64)
・ a single component of layer
and the layer
of the transfer target type
(65)
resp.
(66)
Theorem 21.3 provided the scaffold of the pruned descendant tree
of
, for
,
with mainlines and periodic bifurcations.
With respect to number theoretic applications, however, the following Corollaries 21.2 and 21.3 are of the greatest importance.
Corollary 21.2 Let
be a non-negative integer.
Whereas the vertices with even superscript
,
, that is,
, are merely
links in the distinguished path, the vertices with odd superscript
,
, that is,
, reveal the essential periodic bifurcations with the following properties.
1) The regular component
of the descendant tree
is a coclass tree which
contains the mainline,
![]()
which entirely consists of
-groups, and three distinguished periodic sequences whose vertices
![]()
are
-groups exactly for even
and are characterized by the following TKTs ![]()
with layer
given by
(67)
which deviate from the mainline TKT of Equation (58) in a single component only.
2) The irregular component
of the descendant tree
is a forest which
contains a bunch of 3 isolated Schur
-groups
![]()
which possess the same TKTs as in Equation (67), and additionally contains the root of the next coclass tree
, where
, whose mainline vertices ![]()
with
share the TKT in Equation (58).
The metabelian 3-groups forming the three distinguished periodic sequences
![]()
of the pruned coclass tree
in Corollary 21.2, for
, belong to the few groups for which all immediate descendants with respect to the parent definition (P4) are known (we did not use this kind of
descendants up to now.) Since all groups in
are of derived length 3, the set of these descen-
dants can be defined in the following way.
Definition 21.2 Let
be a finite metabelian
-group. Then the set of all finite non-metabelian
-groups
whose second derived quotient
is isomorphic to
is called the cover
of
. The subset
consisting of all Schur
-groups in
is called the balanced cover of
.
Corollary 21.3 For
, the group
, which does not have a balanced presentation, possesses a
finite cover of cardinality
and a unique Schur
-group in its balanced cover with
. More precisely, the covers are given explicitly by
(68)
The arrows in Figure 8 and Figure 9 indicate the projections
from all members
of a cover
onto the common metabelianization
, that is, in the sense of the parent definition (P4), from the descendants
onto the parent
.
Proof. (of Theorem 21.3, Corollary 21.2, Corollary 21.3 and Theorem 21.4)
The
-group generation algorithm [8] -[10] , which is implemented in the computational algebra system Magma [13] [43] [44] , was used for constructing the pruned descendant trees
with roots
which were defined as the disjoint union of all pruned coclass trees
of the
descendants
,
, of
as roots, together with
siblings in the
irregular component
, 3 of them Schur
-groups with
and
. Using the strict
periodicity [1] [2] of each pruned coclass tree
, which turned out to be of length 2, the vertical
construction was terminated at nilpotency class 19, considerably deeper than the point where periodicity sets in. The horizontal construction was extended up to coclass 10, where the consumption of CPU time became daunting.
Within the frame of our computations, the periodicity was not restriced to bifurcations only: It seems that the
pruned (or maybe even the entire) descendant trees
are all isomorphic to
as
graphs. This is visualized impressively by Figure 8 and Figure 9, where the following notation (not to be confused with layers) is used
![]()
resp.
![]()
Similarly as in the previous section, the extent to which we constructed the pruned descendant trees suggests the following conjecture.
Conjecture 21.2 Theorem 21.3, Corollary 21.2 and Corollary 21.3 remain true for an arbitrarily large positive integer
, not necessarily bounded by 8.
One-parameter polycyclic pc-presentations for the groups in the first three pruned coclass trees of
are given as follows.
1) For the metabelian vertices of the pruned coclass tree
with class
, that is, starting with
and excluding the root
and its descendant
, by
(69)
2) For the non-metabelian vertices of the pruned coclass tree
with class
, and including
the Schur
-groups, which are siblings of the root, by
(70)
3) For the non-metabelian vertices of the pruned coclass tree
with class
, and including
the Schur
-groups, which are siblings of the root, by
(71)
The parameter
is the nilpotency class of the group, and the parameters
and
determine
・ the location of the group on the descendant tree, and
・ the transfer kernel type (TKT) of the group, as follows:
lies on the mainline (this is the so-called mainline principle) and has TKT c.18,
,
whereas all the other groups belong to periodic sequences or are isolated Schur
-groups:
possesses TKT E.6,
,
and
have TKT H.4,
, and lie outside of the pruned tree,
and
have TKT E.14,
.
In Figure 10, resp. Figure 11, we have drawn the lattice of normal subgroups of
, resp.
.
The upper and lower central series,
,
, of these groups form subgraphs whose relative position justifies the names of these series, as visualized impressively by Figure 10 and Figure 11.
Generators
,
, and
, are carefully selected independently from
individual isomorphism types and placed in locations which illustrate the structure of the groups. Furthermore, the normal lattice of the metabelianization
is also included as a subgraph simply by putting
.
We conclude with a theorem concerning the central series and some fundamental properties of the Schur
- groups which we encountered among all the groups under investigation.
Theorem 21.4
Let
be an integer. There exist exactly 6 pairwise non-isomorphic groups
of order
, class
, coclass
, having fixed derived length 3, such that
1) the factors of their upper central series are given by
![]()
![]()
Figure 10. Normal Lattice and Central Series of
.
2) their second derived group
is central and cyclic of order
.
Furthermore,
・ they are Schur
-groups with automorphism group
of order
,
・ the factors of their lower central series are given by
![]()
Figure 11. Normal Lattice and Central Series of
.
![]()
・ their metabelianization
is of order
, class
and of fixed coclass 2,
・ their biggest metabelian ancestor, that is the
th iterated parent, is given by either
or
.
22. Conclusion
We emphasize that the results of Section 21.2 provide the background for considerably stronger assertions than those made in [7] (which are, however, sufficient already to disprove erroneous claims in [48] [49] ). Firstly, they concern four TKTs E.6, E.14, E.8 and E.9 instead of just TKT E.9, and secondly, they apply to varying odd nilpotency class
instead of just class 5.
Acknowledgements
We gratefully acknowledge that our research is supported by the Austrian Science Fund (FWF): P 26008-N25. We are indebted to the anonymous referees for valuable suggestions improving the exposition and readability.