_{1}

^{*}

Theoretical background and an implementation of the p -group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p -groups.

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.

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

Additionally to their order

An important question is how the descendant tree

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

Since computer aided classifications of finite

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

Ascione’s thesis subject [

The goal of Nebelung’s dissertation [

According to Newman ([

[(P)]

1) the centre

2) the last non-trivial term

3) the last non-trivial term

4) the last non-trivial term

In each case,

In the following, the direction of the canonical projections is selected for all edges. Then, more generally, a vertex

of directed edges from

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

with

Generally, the descendant tree

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-

finite. An infinite pro-

A central finiteness result for infinite pro-

・

・

to

・

The descendant tree

minimal

is called the mainline (or trunk) of the tree.

Further terminology, used in diagrams visualizing finite parts of descendant trees, is explained in

If the descendant tree is a coclass tree

labelled according to the level n, then the finite subtree defined as the difference set

is called the nth branch (or twig) of the tree or also the branch

If all vertices of depth bigger than a given integer

(depth-)pruned branch

connected by the mainline, whose vertices

The periodicity of branches of depth-pruned coclass trees has been proved with analytic methods using zeta functions ([

Theorem 7.1 For any infinite pro-

Proof. The graph isomorphisms of depth-

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

See

Assume that parents of finite

The nuclear rank

・

・ If

・ If

In the last case, a more precise assertion is possible: If G has coclass

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,

In this case,

the coclass increases by

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 directed coclass graphs

Coclass Theorems imply that

is the disjoint union of finitely many coclass trees

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 [

Depending on the prime p, there is an upper bound on the order of groups for which a SmallGroup identifier exists, e.g.

and an irregular immediate descendant, connected by an edge of depth

The ANUPQ package [

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.

The coclass graph

of finite p-groups of coclass 0 does not contain a coclass tree and consists of the trivial group 1 and the cyclic group

The coclass graph

of finite p-groups of coclass 1 consists of the unique coclass tree with root

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

For

However, the coclass tree of

With the aid of kernels and targets of Artin transfer homomorphisms [

The concrete examples

by

For

The 2-groups of maximal class, that is of coclass 1, form three periodic infinite sequences:

・ the dihedral groups,

・ the generalized quaternion groups,

・ the semidihedral groups,

For

In contrast to any bigger coclass

The genesis of the coclass graph

As opposed to

For the prime

For odd primes

component

connecting edges of depth 2 of the irregular immediate descendants of

For

・ Firstly, there are two terminal Schur

・ Secondly, the two groups

・ And, finally, the three groups

Displaying additional information on kernels and targets of Artin transfers [

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

inducing the inversion

It should be pointed out that

Eick, Leedham-Green, Newman and O’Brien ([

For

・ Firstly, there are three leaves

・ Secondly, the group

・ And, finally, the three groups

Here, it should be emphasized that

For

Parameters | Abelianization | Class-2 quotient | Class-3 quotient | Class-4 quotient |
---|---|---|---|---|

This unique tree corresponds to the pro-2 group of the family #59 by Newman and O’Brien ([

Here again,

Since the elementary abelian

been described in the section about coclass 2. The irregular component

subgraph

irregular immediate descendants of

For

・ the groups

・ the five groups

The trees arising from the capable vertices are associated with infinite pro-2 groups by Newman and O’Brien ([

The roots of the coclass trees

siblings.

Seven of these nine top level vertices have been investigated by Benjamin, Lemmermeyer and Snyder ([

Descendant trees with central quotients as parents (P1) are implicit in Hall’s 1940 paper [

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 [

The p-group generation algorithm by Newman [

For a finite p-group G, the lower exponent-p central series (briefly lower p-central series) of

descending series

Since any non-trivial finite p-group

SmallGroups identifier of | Hall Senior classification of | Schur multiplier | 2-rank of | 4-rank of | Maximum of |
---|---|---|---|---|---|

(2) | 2 | 0 | 2 | ||

(2) | 2 | 0 | 2 | ||

(2,2) | 2 | 1 | 3 | ||

(2,2) | 2 | 1 | 3 | ||

(2,2) | 2 | 1 | 3 | ||

(2,2,2) | 2 | 2 | 3 | ||

(2,2,2,2) | 3 | 2 or 3 | 4 |

group 1 has

The complete lower p-central series of

since

For the convenience of the reader and for pointing out the shifted numeration, we recall that the (usual) lower

central series of

recursively by

As above, for any non-trivial finite

nilpotency of

Thus, the complete lower central series of

since

The following rules should be remembered for the exponent-

Let

[(R)]

1)

2)

3) For any

4) For any

We point out that every non-trivial finite

and ending in the trivial group

elementary abelian

Let

We can certainly find a presentation of

where

elements

are contained in

Now we can define the

and the exact sequence

shows that

the

Let us assume now that the assigned finite

conditions

nucleus of

as a subgroup of the

of

As before, let

is a quotient of the

The reason is that there exists an epimorphism

denotes the canonical projection. Consequently, we have

since

epimorphism

In particular, an immediate descendant

of

where

A subgroup

characterization is that

Therefore, the first part of our goal to compile a list of all immediate descendants of

have constructed all allowable subgroups of

where

where

among the immediate descendants.

Two allowable subgroups

are the corresponding immediate descendants of

Such an isomorphism

has the property that

and thus induces an automorphism

according to the rule (R2), each extended automorphism

to be the permutation group generated by all permutations induced by automorphisms of

are precisely the orbits of allowable subgroups under the action of the permutation group

Eventually, our goal to compile a list

when we select a representative

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

・ G is terminal if and only if

・ G is capable if and only if

In the case of capability,

of the corresponding allowable subgroup

For the related phenomenon of multifurcation of a descendant tree at a vertex G with nuclear rank

The

We denote the number of all immediate descendants, resp. immediate descendants of step size

metabelian

as given by actual implementations of the

First, let

・ The group

・ The group

・ The group

descendant numbers

Next, let

・ The group

・ The group

・ One of its immediate descendants, the group

In contrast, groups with abelianization of type

・ The group

・ The group

・ The group

Via the isomorphism

group

can be viewed as the additive analogue of the multiplicative group

of all roots of unity.

Let p be a prime number and G be a finite p-group with presentation

of the G-module

Shafarevich ([

minimal number of generators of the Schur multiplier of G, that is

Boston and Nover ([

for all quotients

Furthermore, Blackhurst (in the appendix On the nucleus of certain p-groups of a paper by Boston, Bush and Hajir [

We conclude this section by giving two examples.

・ A finite p-group G has a balanced presentation

is called a Schur group [

・ A finite p-group G satisfies

and only if it has a non-trivial cyclic Schur multiplier

For searching a particular group in a descendant tree

[(F)]

1) filtering the

2) eliminating a set of certain transfer kernel types (TKTs, see ([

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

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.

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 [

The 2-groups under investigation are three-generator groups with elementary abelian commutator factor group of type

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

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

Just within this subsection, we select a special designation for a TKT [[

Definition 21.1 The transfer kernel type (TKT)

members of the first layer

For brevity, we give 2-logarithms of abelian type invariants in the following theorem and we denote iteration

by formal exponents, for instance,

Theorem 21.1 Let

1) In the descendant tree

of (reverse) directed edges with uniform step size 2 such that

(along the path,

of this path share the following common invariants:

・ the transfer kernel type with layer

first component which is total,

・ the 2-multiplicator rank and the nuclear rank, giving rise to the bifurcation,

・ and the counters of immediate descendants,

determining the local structure of the descendant tree.

2) A few other invariants of the vertices

・ the 2-logarithm of the order, the nilpotency class and the coclass,

・ a single component of layer

In view of forthcoming number theoretic applications, we add the following

Corollary 21.1 Let

1) The regular component

contains a unique periodic sequence whose vertices

rized by a permutation TKT

with a single fixed point

2) The irregular component

unique second coclass tree

with

share the TKT in Equation (44).

Proof. (of Theorem 21.1, Corollary 21.1 and Theorem 21.2)

The

defined as the disjoint union of all pruned coclass trees

[

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

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

Remark 21.2 We must emphasize that the root

only, and that the mainline of the coclass tree

・ its root is not a descendant of

・ the TKT of its vertices

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

2) For the mainline vertices of the coclass tree

3) For the mainline vertices of the coclass tree

Theorem 21.2 For higher coclass

To obtain a presentation for the vertices

periodic sequence whose vertices are characterized by the permutation TKT (49), we must only add the single

relation

Theorem 21.2.

We continue this section with periodic bifurcations in trees of 3-groups, which have been discovered in 2012 and 2013 [

These 3-groups are two-generator groups of coclass at least 2 with elementary abelian commutator quotient of type

The two groups

Denote by

Within this subsection, we make use of special designations for transfer kernel types (TKTs) which were defined generally in [[

We are interested in the unavoidable mainline vertices with TKTs c.18,

We point out that, for instance

Remark 21.3 We choose the following sifting strategy for reducing the entire descendant tree

1) keeping all of the 3 terminal step size-2 descendants, which are exactly the Schur

2) eliminating 2 of the 5 terminal step size-1 descendants having TKT

For brevity, we give 3-logarithms of abelian type invariants in the following theorem and we denote iteration

by formal exponents, for instance,

Theorem 21.3 Let

1) In the descendant tree

of (reverse) directed edges of alternating step sizes 1 and 2 such that

resp. all the vertices with odd superscript

of this path share the following common invariants, respectively:

・ the uniform (w.r.t. i) transfer kernel type, containing a total component

・ the 2-multiplicator rank and the nuclear rank,

resp., giving rise to the bifurcation for odd

・ and the counters of immediate descendants,

resp.

determining the local structure of the descendant tree.

2) A few other invariants of the vertices

・ the 3-logarithm of the order, the nilpotency class and the coclass,

resp.

・ a single component of layer

resp.

Theorem 21.3 provided the scaffold of the pruned descendant tree

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

Whereas the vertices with even superscript

links in the distinguished path, the vertices with odd superscript

1) The regular component

contains the mainline,

which entirely consists of

are

with layer

which deviate from the mainline TKT of Equation (58) in a single component only.

2) The irregular component

contains a bunch of 3 isolated Schur

which possess the same TKTs as in Equation (67), and additionally contains the root of the next coclass tree

with

The metabelian 3-groups forming the three distinguished periodic sequences

of the pruned coclass tree

descendants up to now.) Since all groups in

dants can be defined in the following way.

Definition 21.2 Let

Corollary 21.3 For

finite cover of cardinality

The arrows in

Proof. (of Theorem 21.3, Corollary 21.2, Corollary 21.3 and Theorem 21.4)

The

descendants

irregular component

periodicity [

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

graphs. This is visualized impressively by

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

One-parameter polycyclic pc-presentations for the groups in the first three pruned coclass trees of

1) For the metabelian vertices of the pruned coclass tree

2) For the non-metabelian vertices of the pruned coclass tree

the Schur

3) For the non-metabelian vertices of the pruned coclass tree

the Schur

The parameter

・ the location of the group on the descendant tree, and

・ the transfer kernel type (TKT) of the group, as follows:

whereas all the other groups belong to periodic sequences or are isolated Schur

In

The upper and lower central series,

Generators

individual isomorphism types and placed in locations which illustrate the structure of the groups. Furthermore, the normal lattice of the metabelianization

We conclude with a theorem concerning the central series and some fundamental properties of the Schur

Theorem 21.4

Let

1) the factors of their upper central series are given by

2) their second derived group

Furthermore,

・ they are Schur

・ the factors of their lower central series are given by

・ their metabelianization

・ their biggest metabelian ancestor, that is the

We emphasize that the results of Section 21.2 provide the background for considerably stronger assertions than those made in [

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.

Daniel C.Mayer， (2015) Periodic Bifurcations in Descendant Trees of Finite <i>p</i>-Groups。 Advances in Pure Mathematics，05，162-195. doi: 10.4236/apm.2015.54020