_{1}

^{*}

Based on a general theory of descendant trees of finite
*p*-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator qu-otient, the information content of each coclass subtree with metabelian main-line is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data.

Denote by

• Firstly, the well-known virtual periodicity isomorphisms

• Secondly, evidence is provided of co-periodicity isomorphisms

Together with the coclass theorems of Leedham-Green [

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 [

Then we focus on the tree

Finally, we point out that our theory, together with the investigations of Eick [

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 ( [

in particular,

a non-trivial p-group

Definition 2.1. If G is non-abelian, then the class-

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

according to the isomorphism theorem. The lower central series of

and thus

Definition 2.2. For an assigned finite p-group

The mapping

If the root R is abelian, then all vertices of the tree

of directed subgraphs, where

Definition 2.3. Thus, the components

According to the coclass theorems by Leedham-Green [

Definition 2.4. In the present paper, the focus will lie on finite p-groups with fixed prime

and possibly different integers

Remark 2.1. In §§11 and 12 it will turn out that the coclass trees

In general, we denote a graph

Definition 3.1. Let

then

When

Proposition 3.1. Let

Proof. The bijection

The in-resp. out-degree of a vertex

In particular, if

Proposition 3.2. Let

Proof. Recall that each row of the adjacency matrix A of the tree

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

Definition 4.1. Let

If

In particular, if the sets

Definition 4.2. Let

With respect to applications in other mathematical theories, in particular, algebraic number theory and class field theory, certain properties of the

automorphism group

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

Remark 4.1. Suppose that G is a finite p-group with odd prime p. We point out that 2 divides the order

For a brief description of abelian quotient invariants in logarithmic form, we need the concept of nearly homocyclic p-groups. With an arbitrary prime

Definition 4.4. By the nearly homocyclic abelian 3-group

The following invariants

• The logarithmic order

• The nilpotency class

• The coclass

• The order of the automorphism group

• The action flag

• The transfer kernel type (TKT)

• The transfer target type (TTT)

• The abelian quotient invariants of the first TTT component

• The abelian quotient invariants of the commutator subgroup

• The relation rank

• The nuclear rank

Remark 4.2. Abelian quotient invariants are given in logarithmic notation. The transfer kernel type

The graph theoretic structure of a coclass tree

Definition 5.1. Let

For

Let

denote the nth layer of

The width of the tree is the maximal cardinality of its layers,

Each vertex v of the branch

The length

The depth of a branch

Definition 5.2. Let

Throughout this paper, we assume that both, the depth

Lemma 5.1. Let

Proof. Since

and we have

Lemma 5.2. Let

Proof. Since

Since the implementation of the p-group generation algorithm [

Theorem 5.1. Let

1) If the tree is of depth

2) If the tree is of depth

3) If the tree is of depth

Proof. Put

If

If

If

Remark 5.1. In Theorem 5.1, item (1) is included in item (2), since

Corollary 5.1. Under the same assumptions as in Theorem 5.1, the width of the coclass tree

For assigned small values of the depth

1) If the tree is of depth

2) If the tree is of depth

where n runs from

3) If the tree is of depth

where n runs from

Proof. According to [

where

According to Lemma 5.2, we have

For finding the maximal layer cardinality, the root term

In the special case of

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

where

Independently of being metabelian or non-metabelian, a finite 3-group G of order up to

If we define a mainline as a maximal path of infinitely many directed edges of step size

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

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 ( [

Example 7.1. The metabelian 3-group

The metabelian 3-group

Finally, the metabelian 3-group

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

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

Example 7.2. In the second mainline vertex

The same is true for the second mainline vertex

The unnecessary requirement of coclass stability would eliminate the pre-periods of the trees

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 [

The following first main result of this paper establishes the details of the primitive period of branches of five coclass-4 trees

Theorem 8.1. (Main Theorem on Strict Periodicity Isomorphisms of Branches.)

For each integer

• logarithmic order

• nilpotency class

• order of the automorphism group

• first component of the transfer target type

• commutator subgroup

Consequently, the branches of each tree

Proof. The

Invariants connected with the nilpotency class are not strict and satisfy the following transformation laws: the shift

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

Unfortunately it is much less well known that the entire metabelian skeleton

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

Theorem 8.2. (Main Theorem on Co-Periodicity Isomorphisms of Coclass Trees.)

Let the integer

• logarithmic order

• nilpotency class

• coclass

• order of the automorphism group

• first component of the transfer target type

• commutator subgroup

Proof. The statement for the metabelian skeletons

The established

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

Consequently, all coclass trees

The general graph theoretic and algebraic foundations of the coclass forests

The identification of 3-groups G with coclass

where

Metabelian 3-groups with coclass

where

The flow of our investigations is guided by the present section concerning the remarkable infinite main trunk

Theorem 10.1. (The main trunk.)

1) In the descendant tree

2) The trailing vertex

3) All the other vertices

4) For

Restricted to metabelian groups, the immediate descendant numbers are

All immediate descendants are

Proof. See the dissertation of Nebelung ( [

Remark 10.1. Although the number of metabelian children of step sizes

Corollary 10.1. (All coclass trees with metabelian mainlines.)

The coclass trees of 3-groups G with

1) For even

2) For odd

3) The unique pre-periodic exception is the vertex

Proof. See the dissertation of Nebelung ( [

Although formulated for the particular coclass

with the crucial bifucation from

Proposition 11.1 (Co-periodic standard case.)

The sporadic part

• 13 ^{13} with types F.7, F.11, F.12, F.13,

• 8

• 66

• 179 isolated vertices with

• 23 capable vertices with

Whose children do not belong to

The action flag of all metabelian top vertices with depth

Together with the 6 metabelian roots

Proposition 11.2. (Pre-periodic exception.)

The constitution of the sporadic part ^{9}, resp.

• 88

• 12

whose children do not belong to

• 268 isolated vertices with

• 58 capable vertices with

whose children do not belong to

The distribution of the action flags

Together with the 6 metabelian roots

Theorem 11.1. The coclass-r forest

and

The algebraic invariants for groups with positive action flag

Proof. (of Propositions 11.1, 11.2, and Theorem 11.1) We have computed the sporadic parts

# | dp | dl | Type | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 5, 7 | 0 | 2 | 1^{2} | 6 | 4 | 2^{2} | 21^{3} | b.10^{*} | (0043) | 2^{*} | ||

1 | 6, 9 | 0 | 2 | 1^{2} | 6 | 2 | 32 | 2^{3}1 | b.10^{*} | (0043) | 2 | ||

2 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | d.19^{*} | (0343) | 1 | ||

1 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | d.23^{*} | (0243) | 1 | ||

2 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | d.25^{*} | (0143) | 2 | ||

2 | 6, 9 | 0 | 2 | 1^{2} | 4 | 0 | 32 | 2^{3}1 | F.7 | (3443) | 1^{*} | ||

1 | 6, 9 | 0 | 2 | 1^{2} | 4 | 0 | 32 | 2^{3}1 | F.7 | (3443) | 1^{*} | ||

2 | 6, 9 | 0 | 2 | 1^{2} | 4 | 0 | 32 | 2^{3}1 | F.11 | (1143) | 1^{*} | ||

4 | 6, 9 | 0 | 2 | 1^{2} | 4 | 0 | 32 | 2^{3}1 | F.12 | (1343) | 1^{*} | ||

4 | 6, 9 | 0 | 2 | 1^{2} | 4 | 0 | 32 | 2^{3}1 | F.13 | (3143) | 1^{*} | ||

1 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | G.16 | (1243) | 2 | ||

1 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | G.16 | (1243) | 1 | ||

2 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | G.19 | (2143) | 2 | ||

1 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | G.19 | (2143) | 2 | ||

2 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | H.4 | (3343) | 2 | ||

1 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | H.4 | (3343) | 1 | ||

1 | 7, 10 | 1 | 2 | 1 | 4 | 0 | 32 | 32^{2}1 | G.16r | (1243) | 2^{*} | ||

1 | 7, 10 | 1 | 2 | 1 | 4 | 0 | 32 | 2^{4} | G.16i | (1243) | 1^{*} | ||

2 | 7, 10 | 1 | 2 | 1 | 4 | 0 | 32 | 32^{2}1 | G.19r | (2143) | 2^{*} | ||

1 | 7, 10 | 1 | 2 | 1 | 4 | 0 | 32 | 2^{4} | G.19i | (2143) | 2^{*} | ||

2 | 7, 10 | 1 | 2 | 1 | 4 | 0 | 32 | 32^{2}1 | H.4r | (3343) | 2^{*} | ||

1 | 7, 10 | 1 | 2 | 1 | 4 | 0 | 32 | 2^{4} | H.4i | (3343) | 1^{*} | ||

12 | 7, 10 | 1 | 2 | 1 | 4 | 0 | 32 | G or H | 0 | ||||

10 | 7, 10 | 1 | 2 | 1 | 4 | 0 | 32 | G or H | 0 | ||||

8 | 7, 10 | 1 | 3 | 1 | 4 | 0 | 32 | 2^{3}1 | G or H | 0 | |||

20 | 7, 10 | 1 | 3 | 1 | 4 | 0 | 32 | 2^{3}1 | G or H | 0 | |||

4 | 7, 10 | 1 | 3 | 1 | 4 | 0 | 32 | 2^{3}1 | G.19 | (2143) | 0 | ||

6 | 7, 10 | 1 | 3 | 1 | 4 | 0 | 32 | 2^{3}1 | G or H | 0 |

Propositions 11.1 and 11.2, they all share a common graph theoretic structure with

which give rise to the periodic part of

The metabelian skeleton of both,

Proposition 11.3. (Periodicity and descendant numbers.)

The branches

The graph theoretic structure of the tree is determined uniquely by the numbers

Proof. (of Proposition 11.3) The statements concerning the numbers

4, resp. 6, metabelian vertices with bicyclic centre

5, resp. 6, non-metabelian vertices with

together 21 vertices (10 of them metabelian) in the pre-periodic branch

and the primitive period

6, resp. 6, metabelian vertices with

9, resp. 9, non-metabelian vertices with

together 30 vertices (12 of them metabelian) in branch

4, resp. 8, metabelian vertices with

5, resp. 10, non-metabelian vertices with

together 27 vertices (12 of them metabelian) in branch

The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung ( [

The claim of the virtual periodicity of branches has been proved generally for any coclass tree by du Sautoy [

Theorem 11.2. (Graph theoretic and algebraic invariants.)

The coclass-4 tree

1) The pre-period

2) The cardinality of the irregular branch is

3) The branches

4) The cardinalities of the regular branches are

5) Depth, width, and information content of the tree are given by

The algebraic invariants of the groups represented by vertices forming the pre-period

Remark 11.1. The algebraic information in

Proof. (of Theorem 11.2) According to Proposition 11.3, the logarithmic order of the tree root, respectively of the periodic root, is

Since

Proposition 11.3, the unique capable child of

With the aid of Formula (5.9) in Theorem 5.1, the claims (2) and (4) are consequences of Proposition 11.3:

According to Formula (5.13) in Corollary 5.1, where n runs from

# | dp | dl | Type | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 6, 9 | 0 | 2 | 1^{2} | 6 | 2 | 32 | 2^{3}1 | b.10^{*} | (0043) | 2 | ||

1 | 7, 10 | 0 | 2 | 1^{2} | 6 | 1 | 3^{2} | 32^{2}1 | b.10^{*} | (0043) | 2 | ||

1 | 7, 10 | 1 | 2 | 1^{2} | 5 | 0 | 3^{2} | 32^{2}1 | d.19 | (3043) | 0 | ||

1 | 7, 10 | 1 | 2 | 1^{2} | 5 | 0 | 3^{2} | 32^{2}1 | d.23 | (1043) | 0 | ||

1 | 7, 10 | 1 | 2 | 1^{2} | 5 | 0 | 3^{2} | 32^{2}1 | d.25 | (2043) | 0 | ||

1 | 7, 10 | 1 | 2 | 1 | 5 | 0 | 32 | 32^{2}1 | b.10r | (0043) | 2 | ||

2 | 7, 10 | 1 | 2 | 1 | 5 | 0 | 32 | 32^{2}1 | b.10r | (0043) | 0 | ||

1 | 7, 10 | 1 | 2 | 1 | 5 | 0 | 32 | 2^{4} | b.10i | (0043) | 2 | ||

2 | 7, 10 | 1 | 2 | 1 | 5 | 0 | 32 | 2^{4} | b.10i | (0043) | 0 | ||

3 | 7, 10 | 1 | 3 | 1^{2} | 5 | 0 | 32 | 2^{3}1 | b.10 | (0043) | 0 | ||

2 | 7, 10 | 1 | 3 | 1^{2} | 5 | 0 | 32 | 2^{3}1 | b.10 | (0043) | 0 | ||

2 | 7, 10 | 1 | 3 | 1 | 5 | 0 | 32 | 2^{3}1 | b.10 | (0043) | 0 | ||

2 | 7, 10 | 1 | 3 | 1 | 5 | 0 | 32 | 2^{3}1 | b.10 | (0043) | 0 | ||

2 | 7, 10 | 1 | 3 | 1 | 5 | 0 | 32 | 2^{3}1 | b.10 | (0043) | 0 | ||

1 | 8, 11 | 0 | 2 | 1^{2} | 6 | 1 | 43 | 3^{2}21 | b.10^{*} | (0043) | 2 | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 5 | 0 | 43 | 3^{2}21 | d.19 | (3043) | 1 | ||

1 | 8, 11 | 1 | 2 | 1^{2} | 5 | 0 | 43 | 3^{2}21 | d.23 | (1043) | 1 | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 5 | 0 | 43 | 3^{2}21 | d.25 | (2043) | 2 | ||

3 | 8, 11 | 1 | 2 | 1 | 5 | 0 | 3^{2} | 3^{2}21 | b.10 | (0043) | 0 | ||

3 | 8, 11 | 1 | 2 | 1 | 5 | 0 | 3^{2} | 3^{2}21 | b.10 | (0043) | 0 | ||

6 | 8, 11 | 1 | 3 | 1^{2} | 5 | 0 | 3^{2} | 32^{2}1 | b.10 | (0043) | 1 | ||

2 | 8, 11 | 1 | 3 | 1^{2} | 5 | 0 | 3^{2} | 32^{2}1 | b.10 | (0043) | 2 | ||

1 | 8, 11 | 1 | 3 | 1^{2} | 5 | 0 | 3^{2} | 32^{2}1 | b.10 | (0043) | 1 | ||

6 | 8, 11 | 1 | 3 | 1 | 5 | 0 | 3^{2} | 32^{2}1 | b.10 | (0043) | 0 | ||

2 | 8, 11 | 1 | 3 | 1 | 5 | 0 | 3^{2} | 32^{2}1 | b.10 | (0043) | 0 | ||

1 | 8, 11 | 1 | 3 | 1 | 5 | 0 | 3^{2} | 32^{2}1 | b.10 | (0043) | 0 | ||

1 | 9, 12 | 0 | 2 | 1^{2} | 6 | 1 | 4^{2} | 4321 | b.10^{*} | (0043) | 2 | ||

1 | 9, 12 | 1 | 2 | 1^{2} | 5 | 0 | 4^{2} | 4321 | d.19 | (3043) | 0 | ||

1 | 9, 12 | 1 | 2 | 1^{2} | 5 | 0 | 4^{2} | 4321 | d.23 | (1043) | 0 | ||

1 | 9, 12 | 1 | 2 | 1^{2} | 5 | 0 | 4^{2} | 4321 | d.25 | (2043) | 0 | ||

2 | 9, 12 | 1 | 2 | 1 | 5 | 0 | 43 | 4321 | b.10 | (0043) | 2 | ||

4 | 9, 12 | 1 | 2 | 1 | 5 | 0 | 43 | 4321 | b.10 | (0043) | 0 | ||

2 | 9, 12 | 1 | 2 | 1 | 5 | 0 | 43 | 4321 | b.10 | (0043) | 0 | ||

3 | 9, 12 | 1 | 3 | 1^{2} | 5 | 0 | 43 | 3^{2}21 | b.10 | (0043) | 0 | ||

2 | 9, 12 | 1 | 3 | 1^{2} | 5 | 0 | 43 | 3^{2}21 | b.10 | (0043) | 0 | ||

6 | 9, 12 | 1 | 3 | 1 | 5 | 0 | 43 | 3^{2}21 | b.10 | (0043) | 0 | ||

4 | 9, 12 | 1 | 3 | 1 | 5 | 0 | 43 | 3^{2}21 | b.10 | (0043) | 0 |

The information content of the tree is given by Formula (5.17) in the Definition 5.3:

The algebraic invariants in

Each group is characterized by the parameters of the normalized representative

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

1) The groups with

2) With respect to the kernel types, all mainline groups of type b.10^{*},

3) The relation rank is given by

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 [

Proposition 11.4. (Periodicity and descendant numbers.)

The branches ^{*},

The graph theoretic structure of the tree is determined uniquely by the numbers

Proof. (of Proposition 11.4) The statements concerning the numbers

5, resp. 9, metabelian vertices with

8, resp. 16, non-metabelian vertices with

( children of , and children of with depth 1)

together 38 vertices (14 of them metabelian) in branch

9, resp. 10, metabelian vertices with

16, resp. 16, non-metabelian vertices with

( 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 ( [

The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [

Theorem 11.3. (Graph theoretic and algebraic invariants.)

The coclass-4 tree

1) The branches

2) The cardinalities of the periodic branches are

3) Depth, width, and information content of the tree are given by

The algebraic invariants of the vertices forming the primitive period

Proof. (of Theorem 11.3) Since every mainline vertex

# | dp | dl | Type | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | d.19^{*} | (0343) | 1 | ||

1 | 7, 10 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{2}1 | d.19^{*} | (0343) | 1^{*} | ||

1 | 7, 10 | 1 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | F.7 | (4343) | 0 | 3^{16} | |

1 | 7, 10 | 1 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | F.12 | (1343) | 0 | 3^{16} | |

1 | 7, 10 | 1 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | F.13 | (2343) | 0 | 3^{16} | |

1 | 7, 10 | 1 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{2}1 | H.4 | (3343) | 0 | 3^{16} | |

6 | 7, 10 | 1 | 3 | 1^{2} | 4 | 0 | 32 | 2^{3}1 | d.19 | (0343) | 0 | 3^{15} | |

2 | 7, 10 | 1 | 3 | 1^{2} | 4 | 0 | 32 | 2^{3}1 | d.19 | (0343) | 0 | 3^{14} | |

9 | 8, 11 | 2 | 2 | 1 | 4 | 0 | 3^{2} | 3^{2}21 | H.4 | (3343) | 0 | 3^{18} | |

12 | 8, 11 | 2 | 3 | 1 | 4 | 0 | 3^{2} | 32^{2}1 | H.4 | (3343) | 0 | 3^{17} | |

4 | 8, 11 | 2 | 3 | 1 | 4 | 0 | 3^{2} | 32^{2}1 | H.4 | (3343) | 0 | 3^{16} | |

1 | 8, 11 | 0 | 2 | 1^{2} | 5 | 1 | 43 | 3^{2}21 | d.19^{*} | (0343) | 1 | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 3^{2}21 | F.7 | (4343) | 1^{*} | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 3^{2}21 | F.12 | (1343) | 1^{*} | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 3^{2}21 | F.13 | (2343) | 1^{*} | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 5 | 1 | 43 | 3^{2}21 | H.4 | (3343) | 1 | ||

12 | 8, 11 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | d.19 | (0343) | 1^{*} | ||

4 | 8, 11 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | d.19 | (0343) | 1^{*} | ||

2 | 9, 12 | 2 | 2 | 1 | 4 | 0 | 43 | 4321 | H.4 | (3343) | 1^{*} | ||

8 | 9, 12 | 2 | 2 | 1 | 4 | 0 | 43 | 4321 | H.4 | (3343) | 0 | 3^{20} | |

12 | 9, 12 | 2 | 3 | 1 | 4 | 0 | 43 | 3^{2}21 | H.4 | (3343) | 0 | 3^{19} | |

4 | 9, 12 | 2 | 3 | 1 | 4 | 0 | 43 | 3^{2}21 | H.4 | (3343) | 0 | 3^{18} |

Applied to the primitive period, this yields

taken over all branch roots

Finally, we have

Corollary 11.2. (Actions and relation ranks.) The algebraic invariants of the vertices of the structured coclass-4 tree

1) There are no groups with

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

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 [

Theorem 11.4. (Strict isomorphism of the two trees.)

Viewed as an algebraically structured infinite digraph, the second coclass-4 tree ^{*} in ^{*} in

Proof. (Proof of Theorem 11.3 and Theorem 11.4) The claims have been verified with the aid of MAGMA [

Proposition 11.5. (A special nearly strict isomorphism.)

Viewed as an algebraically structured infinite digraph, the unique coclass-4 tree ^{*} in ^{*} in ^{*} in

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

Proof. This follows immediately from comparing

Proposition 11.6. (Periodicity and descendant numbers.)

The branches

^{*},

The graph theoretic structure of the tree is determined uniquely by the numbers

# | dp | dl | Type | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 32 | 2^{3}1 | d.23* | (0243) | 1 | ||

1 | 7, 10 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{2}1 | d.23* | (0243) | 1^{*} | ||

1 | 7, 10 | 1 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | F.11 | (2243) | 0 | 3^{16} | |

1 | 7, 10 | 1 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | F.12 | (3243) | 0 | 3^{16} | |

1 | 7, 10 | 1 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | F.12 | (4243) | 0 | 3^{16} | |

1 | 7, 10 | 1 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{2}1 | G.16 | (1243) | 0 | 3^{16} | |

6 | 7, 10 | 1 | 3 | 1^{2} | 4 | 0 | 32 | 2^{3}1 | d.23 | (0243) | 0 | 3^{15} | |

2 | 7, 10 | 1 | 3 | 1^{2} | 4 | 0 | 32 | 2^{3}1 | d.23 | (0243) | 0 | 3^{14} | |

9 | 8, 11 | 2 | 2 | 1 | 4 | 0 | 3^{2} | 3^{2}21 | G.16 | (1243) | 0 | 3^{18} | |

12 | 8, 11 | 2 | 3 | 1 | 4 | 0 | 3^{2} | 32^{2}1 | G.16 | (1243) | 0 | 3^{17} | |

4 | 8, 11 | 2 | 3 | 1 | 4 | 0 | 3^{2} | 32^{2}1 | G.16 | (1243) | 0 | 3^{16} | |

1 | 8, 11 | 0 | 2 | 1^{2} | 5 | 1 | 43 | 3^{2}21 | d.23* | (0243) | 1 | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 3^{2}21 | F.11 | (2243) | 1^{*} | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 3^{2}21 | F.12 | (3243) | 1^{*} | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 3^{2}21 | F.12 | (4243) | 1^{*} | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 5 | 1 | 43 | 3^{2}21 | G.16 | (1243) | 1 | ||

12 | 8, 11 | 1 | 3 | 1^{2} | 4 | 0 | 32 | 32^{2}1 | d.23 | (0243) | 1^{*} | ||

4 | 8, 11 | 1 | 3 | 1^{2} | 4 | 0 | 32 | 32^{2}1 | d.23 | (0243) | 1^{*} | ||

2 | 9, 12 | 2 | 2 | 1 | 4 | 0 | 43 | 4321 | G.16 | (1243) | 1^{*} | ||

8 | 9, 12 | 2 | 2 | 1 | 4 | 0 | 43 | 4321 | G.16 | (1243) | 0 | 3^{20} | |

12 | 9, 12 | 2 | 3 | 1 | 4 | 0 | 43 | 3^{2}21 | G.16 | (1243) | 0 | 3^{19} | |

4 | 9, 12 | 2 | 3 | 1 | 4 | 0 | 43 | 3^{2}21 | G.16 | (1243) | 0 | 3^{18} |

Proof. This is a consequence of Proposition 11.5 together with Proposition 11.4. The tree

Theorem 11.5. (Graph theoretic and algebraic invariants.)

The coclass-4 tree

1) The branches

2) The cardinalities of the periodic branches are

3) Depth, width, and information content of the tree are given by

The algebraic invariants of the vertices forming the primitive period

Proof. Since the tree

Corollary 11.3. (Actions and relation ranks.) The algebraic invariants of the vertices of the structured coclass-4 tree

1) There are no groups with

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

Proposition 11.7. (Periodicity and descendant numbers.)

The branches^{*},

The structure of the tree is determined uniquely by the numbers

Theorem 11.6. (Graph theoretic and algebraic invariants.)

The coclass-4 tree

1) The branches

2) The cardinalities of the periodic branches are

3) Depth, width, and information content of the tree are given by

The algebraic invariants of the vertices forming the primitive period

# | dp | dl | Type | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 6, 9 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 2^{3}1 | d.25* | (0143) | 2 | ||

1 | 7, 10 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{2}1 | d.25* | (0143) | 2^{*} | ||

1 | 7, 10 | 1 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | F.11 | (1143) | 0 | 3^{16} | |

1 | 7, 10 | 1 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | F.13 | (3143) | 0 | 3^{16} | |

1 | 7, 10 | 1 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{2}1 | G.19 | (2143) | 0 | 3^{16} | |

3 | 7, 10 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 2^{3}1 | d.25 | (0143) | 0 | 3^{15} | |

2 | 7, 10 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 2^{3}1 | d.25 | (0143) | 0 | 3^{14} | |

6 | 8, 11 | 2 | 2 | 1 | 4 | 0 | 3^{2} | 3^{2}21 | G.19 | (2143) | 0 | 3^{18} | |

7 | 8, 11 | 2 | 3 | 1 | 4 | 0 | 3^{2} | 32^{2}1 | G.19 | (2143) | 0 | 3^{17} | |

2 | 8, 11 | 2 | 3 | 1 | 4 | 0 | 3^{2} | 32^{2}1 | G.19 | (2143) | 0 | 3^{16} | |

1 | 8, 11 | 0 | 2 | 1^{2} | 5 | 1 | 43 | 3^{2}21 | d.25* | (0143) | 2 | ||

1 | 8, 11 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 3^{2}21 | F.11 | (1143) | 1^{*} | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 3^{2}21 | F.13 | (3143) | 1^{*} | ||

2 | 8, 11 | 1 | 2 | 1^{2} | 5 | 1 | 43 | 3^{2}21 | G.19 | (2143) | 2 | ||

7 | 8, 11 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | d.25 | (0143) | 1^{*} | ||

2 | 8, 11 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 32^{2}1 | d.25 | (0143) | 2^{*} | ||

2 | 9, 12 | 2 | 2 | 1 | 4 | 0 | 43 | 4321 | G.19 | (2143) | 2^{*} | ||

6 | 9, 12 | 2 | 2 | 1 | 4 | 0 | 43 | 4321 | G.19 | (2143) | 0 | 3^{20} | |

8 | 9, 12 | 2 | 3 | 1 | 4 | 0 | 43 | 3^{2}21 | G.19 | (2143) | 0 | 3^{19} | |

2 | 9, 12 | 2 | 3 | 1 | 4 | 0 | 43 | 3^{2}21 | G.19 | (2143) | 0 | 3^{18} |

Corollary 11.4. (Actions and relation ranks.)

The algebraic invariants of the vertices of the structured coclass-4 tree

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

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

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,

4, resp. 6, metabelian vertices with

5, resp. 9, non-metabelian vertices with

(children of, and children of with depth 1)

together 24 vertices (10 of them metabelian) in branch

6, resp. 8, metabelian vertices with

9, resp. 10, non-metabelian vertices with

(children of, and children of, both with depth 1)

together 33 vertices (14 of them metabelian) in branch

The tree

Theorem 11.7. (Strict isomorphism of the two trees.)

Viewed as an algebraically structured infinite digraph, the second coclass-4 tree ^{*} in ^{*} in

Proof. (Proof of Thm. 11.6 and Thm 11.7.) The claims have been verified with the aid of MAGMA [

Although formulated for the particular coclass

Proposition 12.1. The sporadic part

• 7

• 4

• 34

• 89 isolated vertices with

• 13 capable vertices with

whose children do not belong to

The action flag of all vertices is

Together with the 4 metabelian roots of coclass-5 trees, the

Theorem 12.1. The coclass-r forest

and

Proof. (of Proposition 12.1 and Theorem 12.1) We have computed the sporadic parts

which give rise to the periodic part of

children, there are

The metabelian skeleton consists of

Proposition 12.2. (Periodicity and descendant numbers.)

# | dp | dl | Type | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 6, 9 | 0 | 2 | 1^{2} | 6 | 2 | 3^{2} | 2^{3}1 | b.10^{*} | (043) | 2 | ||

1 | 7, 11 | 0 | 2 | 1^{2} | 6 | 2 | 3^{2} | 32^{3} | b.10^{*} | (0043) | 2 | ||

1 | 7, 11 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{3} | d.19^{*} | (0343) | 0 | ||

1 | 7, 11 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{3} | d.23^{*} | (0243) | 0 | ||

1 | 7, 11 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{3} | d.25^{*} | (0143) | 0 | ||

1 | 7, 11 | 0 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{3} | F.7 | (3443) | 0 | ||

2 | 7, 11 | 0 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{3} | F.11 | (1143) | 0 | ||

2 | 7, 11 | 0 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{3} | F.12 | (1343) | 0 | ||

2 | 7, 11 | 0 | 2 | 1^{2} | 4 | 0 | 3^{2} | 32^{3} | F.13 | (3143) | 0 | ||

1 | 7, 11 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{3} | G.16 | (1243) | 0 | ||

1 | 7, 11 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{3} | G.19 | (2143) | 0 | ||

2 | 7, 11 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{3} | H.4 | (3343) | 0 | ||

12 | 8, 12 | 1 | 2 | 1 | 4 | 0 | 3^{2} | 3^{2}2^{2} | 0 | ||||

12 | 8, 12 | 1 | 2 | 1 | 4 | 0 | 3^{2} | 3^{2}2^{2} | 0 | ||||

8 | 8, 12 | 1 | 3 | 1 | 4 | 0 | 3^{2} | 32^{3} | 0 | ||||

20 | 8, 12 | 1 | 3 | 1 | 4 | 0 | 3^{2} | 32^{3} | 0 | ||||

8 | 8, 12 | 1 | 3 | 1 | 4 | 0 | 3^{2} | 32^{3} | 0 |

The branches^{*},

The graph theoretic structure of the tree is determined uniquely by the numbers

Proof. (of Proposition 12.2) The statements concerning the numbers

6, resp. 6, metabelian vertices with bicyclic centre

9, resp. 9, non-metabelian vertices with

together 30 vertices (12 of them metabelian) in the pre-periodic branch

and the primitive period

4, resp. 6, metabelian vertices with

5, resp. 9, non-metabelian vertices with

together 24 vertices (10 of them metabelian) in branch

6, resp. 9, metabelian vertices with

9, resp. 16, non-metabelian vertices with

together 40 vertices (15 of them metabelian) in branch

The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung ( [

The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [

Theorem 12.2. (Graph theoretic and algebraic invariants.)

The coclass-5 tree

1) The pre-period

2) The cardinality of the irregular branch is

3) The branches

4) The cardinalities of the regular branches are

5) Depth, width, and information content of the tree are given by

The algebraic invariants of the groups represented by vertices forming the pre-period

Remark 12.1. The algebraic information in

Proof. (of Theorem 12.2) According to Proposition 12.2, the logarithmic order of the tree root, respectively of the periodic root, is

Since

# | dp | dl | Type | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 7, 11 | 0 | 2 | 1^{2} | 6 | 2 | 3^{2} | 32^{3} | b.10* | (0043) | 2^{*} | ||

1 | 8, 12 | 0 | 2 | 1^{2} | 6 | 1 | 43 | 3^{2}2^{2} | b.10* | (0043) | 2 | ||

2 | 8, 12 | 1 | 2 | 1^{2} | 5 | 0 | 43 | 3^{2}2^{2} | d.19 | (3043) | 1^{*} | ||

1 | 8, 12 | 1 | 2 | 1^{2} | 5 | 0 | 43 | 3^{2}2^{2} | d.23 | (1043) | 1^{*} | ||

2 | 8, 12 | 1 | 2 | 1^{2} | 5 | 0 | 43 | 3^{2}2^{2} | d.25 | (2043) | 2^{*} | ||

3 | 8, 12 | 1 | 2 | 1 | 5 | 0 | 3^{2} | 3^{2}2^{2} | b.10 | (0043) | 2^{*} | ||

3 | 8, 12 | 1 | 2 | 1 | 5 | 0 | 3^{2} | 3^{2}2^{2} | b.10 | (0043) | 1^{*} | ||

6 | 8, 12 | 1 | 3 | 1^{2} | 5 | 0 | 3^{2} | 32^{3} | b.10 | (0043) | 1^{*} | ||

2 | 8, 12 | 1 | 3 | 1^{2} | 5 | 0 | 3^{2} | 32^{3} | b.10 | (0043) | 2^{*} | ||

1 | 8, 12 | 1 | 3 | 1^{2} | 5 | 0 | 3^{2} | 32^{3} | b.10 | (0043) | 1^{*} | ||

2 | 8, 12 | 1 | 3 | 1 | 5 | 0 | 3^{2} | 32^{3} | b.10 | (0043) | 2^{*} | ||

5 | 8, 12 | 1 | 3 | 1 | 5 | 0 | 3^{2} | 32^{3} | b.10 | (0043) | 1^{*} | ||

2 | 8, 12 | 1 | 3 | 1 | 5 | 0 | 3^{2} | 32^{3} | b.10 | (0043) | 1^{*} | ||

1 | 9, 13 | 0 | 2 | 1^{2} | 6 | 1 | 4^{2} | 432^{2} | b.10* | (0043) | 2^{*} | ||

1 | 9, 13 | 1 | 2 | 1^{2} | 5 | 0 | 4^{2} | 432^{2} | d.19 | (3043) | 0 | ||

1 | 9, 13 | 1 | 2 | 1^{2} | 5 | 0 | 4^{2} | 432^{2} | d.23 | (1043) | 0 | ||

1 | 9, 13 | 1 | 2 | 1^{2} | 5 | 0 | 4^{2} | 432^{2} | d.25 | (2043) | 0 | ||

3 | 9, 13 | 1 | 2 | 1 | 5 | 0 | 43 | 432^{2} | b.10 | (0043) | 0 | ||

3 | 9, 13 | 1 | 2 | 1 | 5 | 0 | 43 | 432^{2} | b.10 | (0043) | 0 | ||

3 | 9, 13 | 1 | 3 | 1^{2} | 5 | 0 | 43 | 3^{2}2^{2} | b.10 | (0043) | 0 | ||

2 | 9, 13 | 1 | 3 | 1^{2} | 5 | 0 | 43 | 3^{2}2^{2} | b.10 | (0043) | 0 | ||

6 | 9, 13 | 1 | 3 | 1 | 5 | 0 | 43 | 3^{2}2^{2} | b.10 | (0043) | 0 | ||

2 | 9, 13 | 1 | 3 | 1 | 5 | 0 | 43 | 3^{2}2^{2} | b.10 | (0043) | 0 | ||

1 | 9, 13 | 1 | 3 | 1 | 5 | 0 | 43 | 3^{2}2^{2} | b.10 | (2043) | 0 | ||

1 | 10, 14 | 0 | 2 | 1^{2} | 6 | 1 | 54 | 4^{2}2^{2} | b.10* | (0043) | 2 | ||

2 | 10, 14 | 1 | 2 | 1^{2} | 5 | 0 | 54 | 4^{2}2^{2} | d.19 | (3043) | 1^{*} | ||

1 | 10, 14 | 1 | 2 | 1^{2} | 5 | 0 | 54 | 4^{2}2^{2} | d.23 | (0043) | 1^{*} | ||

2 | 10, 14 | 1 | 2 | 1^{2} | 5 | 0 | 54 | 4^{2}2^{2} | d.25 | (0043) | 2^{*} | ||

9 | 10, 14 | 1 | 2 | 1 | 5 | 0 | 4^{2} | 4^{2}2^{2} | b.10 | (0043) | 1^{*} | ||

6 | 10, 14 | 1 | 3 | 1^{2} | 5 | 0 | 4^{2} | 432^{2} | b.10 | (0043) | 1^{*} | ||

2 | 10, 14 | 1 | 3 | 1^{2} | 5 | 0 | 4^{2} | 432^{2} | b.10 | (0043) | 2^{*} | ||

1 | 10, 14 | 1 | 3 | 1^{2} | 5 | 0 | 4^{2} | 432^{2} | b.10 | (0043) | 1^{*} | ||

12 | 10, 14 | 1 | 3 | 1 | 5 | 0 | 4^{2} | 432^{2} | b.10 | (0043) | 1^{*} | ||

4 | 10, 14 | 1 | 3 | 1 | 5 | 0 | 4^{2} | 432^{2} | b.10 | (0043) | 1^{*} |

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

The information content of the tree is given by Formula (5.17) in the Definition 5.3:

The algebraic invariants in

Each group is characterized by the parameters of the normalized repre- sentative

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

1) The groups with

2) With respect to the kernel types, all mainline groups of type b.10^{*},

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

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 [

Proposition 12.3. (Periodicity and descendant numbers.)

The branches ^{*},

The graph theoretic structure of the tree is determined uniquely by the numbers

Proof. (of Proposition 12.3) The statements concerning the numbers

9, resp. 18, metabelian vertices with

16, resp. 32, non-metabelian vertices with

(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 ( [

The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [

Theorem 12.3. (Graph theoretic and algebraic invariants.)

The coclass-5 tree

1) The branches

2) The cardinality of the periodic branch is

3) Depth, width, and information content of the tree are given by

The algebraic invariants of the vertices forming the root and the primitive period

Proof. (Proof of Theorem 12.3) Since every mainline vertex

# | dp | dl | Type | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 7, 11 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{3} | d.19^{*} | (0343) | 0 | ||

1 | 8, 12 | 0 | 2 | 1^{2} | 5 | 1 | 43 | 3^{2}2^{2} | d.19^{*} | (0343) | 0 | ||

2 | 8, 12 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 32^{2}1 | F.7 | (4343) | 0 | ||

2 | 8, 12 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 32^{2}1 | F.12 | (1343) | 0 | ||

2 | 8, 12 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 32^{2}1 | F.13 | (2343) | 0 | ||

2 | 8, 12 | 1 | 2 | 1^{2} | 2 | 1 | 43 | 32^{2}1 | H.4 | (3343) | 0 | ||

12 | 8, 12 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 32^{3} | d.19 | (0343) | 0 | ||

4 | 8, 12 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 32^{3} | d.19 | (0343) | 0 | ||

9 | 9, 13 | 2 | 2 | 1 | 4 | 0 | 43 | 432^{2} | H.4 | (3343) | 0 | ||

12 | 9, 13 | 2 | 3 | 1 | 4 | 0 | 43 | 3^{2}2^{2} | H.4 | (3343) | 0 | ||

4 | 9, 13 | 2 | 3 | 1 | 4 | 0 | 43 | 3^{2}2^{2} | H.4 | (3343) | 0 |

the sum of the number

Applied to the primitive period, this yields

taken over all branch roots

Finally, we have

Corollary 12.2. (Actions and relation ranks.)

The algebraic invariants of the vertices of the structured coclass-5 tree

1) There are no groups with GI-action, let alone with RI- or

2) The relation rank is given by

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 [

Proposition 12.4. (Periodicity and descendant numbers.)

The branches ^{*},

The graph theoretic structure of the tree is determined uniquely by the numbers

Proof. (of Proposition 12.4) The statements concerning the numbers

6, resp. 9, metabelian vertices with

9, resp. 16, non-metabelian vertices with

(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 ( [

The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [

Theorem 12.4. (Graph theoretic and algebraic invariants.)

The coclass-5 tree

1) The branches

2) The cardinality of the periodic branch is

3) Depth, width, and information content of the tree are given by

The algebraic invariants of the vertices forming the root and the primitive period

# | dp | dl | Type | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 7, 11 | 0 | 2 | 1^{2} | 5 | 1 | 3^{2} | 32^{3} | d.23^{*} | (0243) | 0 | ||

1 | 8, 12 | 0 | 2 | 1^{2} | 5 | 1 | 43 | 3^{2}2^{2} | d.23^{*} | (0243) | 0 | ||

2 | 8, 12 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 3^{2}2^{2} | F.11 | (2243) | 0 | ||

2 | 8, 12 | 1 | 2 | 1^{2} | 4 | 0 | 43 | 3^{2}2^{2} | F.12 | (3243) | 0 | ||

1 | 8, 12 | 1 | 2 | 1^{2} | 5 | 1 | 43 | 3^{2}2^{2} | G.16 | (1243) | 0 | ||

6 | 8, 12 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 32^{3} | d.23 | (0243) | 0 | ||

2 | 8, 12 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 32^{3} | d.23 | (0243) | 0 | ||

1 | 8, 12 | 1 | 3 | 1^{2} | 4 | 0 | 3^{2} | 32^{3} | d.23 | (0243) | 0 | ||

9 | 9, 13 | 2 | 2 | 1 | 4 | 0 | 43 | 432^{2} | G.16 | (1243) | 0 | ||

12 | 9, 13 | 2 | 3 | 1 | 4 | 0 | 43 | 3^{2}2^{2} | G.16 | (1243) | 0 | ||

4 | 9, 13 | 2 | 3 | 1 | 4 | 0 | 43 | 3^{2}2^{2} | G.16 | (1243) | 0 |

Proof. (Proof of Theorem 12.4) Since every mainline vertex

Applied to the primitive period, this yields

taken over all branch roots

Corollary 12.3. (Actions and relation ranks.) The algebraic invariants of the vertices of the structured coclass-5 tree

1) There are no groups with GI-action, let alone with RI- or

2) The relation rank is given by

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 [

Proposition 12.5. (Periodicity and descendant numbers.)

The branches ^{*},

The graph theoretic structure of the tree is determined uniquely by the numbers

Proof. (of Proposition 12.5) The statements concerning the numbers

6, resp. 12, metabelian vertices with

9, resp. 18, non-metabelian vertices with

(i.e. children of , and children of , both with depth 1)

together 45 vertices (18 of them metabelian) in branch

6, resp. 9, metabelian vertices with

9, resp. 16, non-metabelian vertices with

(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 ( [

The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [

Theorem 12.5. (Graph theoretic and algebraic invariants.)

The coclass-5 tree

1) The branches

2) The cardinalities of the periodic branches are

3) Depth, width, and information content of the tree are given by

The algebraic invariants of the vertices forming the root and the primitive period

Proof. (Proof of Theorem 12.5) Since every mainline vertex