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, 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.

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.

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

of directed edges from to. The vertices forming the path necessarily coincide with the iterated parents of, with:

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:

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.

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

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

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

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.

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

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.

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

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.

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

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

of directed coclass graphs, which are rather forests than trees. More precisely, the above mentioned

Coclass Theorems imply that

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.

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] .

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.

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] .

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.

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

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

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

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

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

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,

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.

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

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

Now we can define the -covering group of by

and the exact sequence

shows that is an extension of by the elementary abelian -multiplicator. We call

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

as a subgroup of the -multiplicator. Consequently, the nuclear rank

of is bounded from above by the -multiplicator rank.

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

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

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

where will be redundant, due to isomorphisms

among the immediate descendants.

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

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.

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

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).

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

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.

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 as in the previous section. Then the second cohomology group

of the G-module is called the Schur multiplier of G. It can also be interpreted as the quotient group

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

Boston and Nover ([39] , 3.2, Prop. 2) have shown that

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,

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.

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.

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] .

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.

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