Periodic bifurcations in descendant trees of finite (p)-groups

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.


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 discovered periodic patterns in descendant trees with promising arithmetical applications form the topic of the final § 21 and the coronation of the entire work.

The structure: 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 p n , for a fixed prime number p and varying integer exponents n ≥ 0. Such groups are briefly called finite p-groups. The vertices of a descendant tree are isomorphism classes of finite p-groups.
Additionally to their order p n , finite p-groups possess two further related invariants, the nilpotency class c and the coclass r := n − c ( § § 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 r, reveal a repeating finite pattern ( § 7). These two crucial properties of finiteness and periodicity, which have been proved independently by M. du Sautoy [1] and by B. Eick and C.R. 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], descendant trees can be endowed with additional structure [4,5,6], which recently turned out to be decisive for arithmetical 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 T (R) can actually be constructed for an assigned starting group which is taken as the root R of the tree. Sections § § 13 -19 are devoted to recall a minimum of the necessary background concerning the p-group generation algorithm by M.F. Newman [8] and E.A. 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, According to M.F. Newman [20, § 2, pp.52-53], there exist several distinct definitions of the parent π(G) of a finite p-group G. The common principle is to form the quotient π(G) := G/N of G by a suitable normal subgroup N G which can be either (P1) the centre N = ζ 1 (G) of G, whence π(G) = G/ζ 1 (G) is called central quotient of G or (P2) the last non-trivial term N = γ c (G) of the lower central series of G, where c denotes the nilpotency class of G or (P3) the last non-trivial term N = P c−1 (G) of the lower exponent-p central series of G, where c denotes the exponent-p class of G or (P4) the last non-trivial term N = G (d−1) of the derived series of G, where d denotes the derived length of G. In each case, G is called an immediate descendant of π(G) and a directed edge of the tree is defined either by G → π(G) in the direction of the canonical projection π : G → π(G) onto the quotient π(G) = G/N or by π(G) → G 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 D. Segal [22, § 7, p In the following, the direction of the canonical projections is selected for all edges. Then, more generally, a vertex R is a descendant of a vertex P , and P is an ancestor of R, if either R is equal to P or there is a path (4.1) R = Q 0 → Q 1 → · · · → Q m−1 → Q m = P, with m ≥ 1, of directed edges from R to P . The vertices forming the path necessarily coincide with the iterated parents Q j = π j (R) of R, with 0 ≤ j ≤ m: (4.2) R = π 0 (R) → π 1 (R) → · · · → π m−1 (R) → π m (R) = P, with m ≥ 1.
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 R/γ c+1−j (R) of class c − j of R when the nilpotency class of R is given by c ≥ m: with c ≥ m ≥ 1. Generally, the descendant tree T (G) of a vertex G is the subtree of all descendants of G, starting at the root G. The maximal possible descendant tree T (1) of the trivial group 1 contains all finite p-groups and is somewhat exceptional, since, for any parent definition (P1-P4), the trivial group 1 has infinitely many abelian p-groups as its immediate descendants. The parent definitions (P2-P3) have the advantage that any non-trivial finite p-group (of order divisible by p) possesses only finitely many immediate descendants.

Pro-p groups and coclass trees
For a sound understanding of coclass trees as a particular instance of descendant trees, it is necessary to summarize some facts concerning infinite topological pro-p groups. The members γ j (S), with j ≥ 1, of the lower central series of a pro-p group S are open and closed subgroups of finite index, and therefore the corresponding quotients S/γ j (S) are finite p-groups. The pro-p group S is said to be of coclass cc(S) := r when the limit r = lim j→∞ cc(S/γ j (S)) of the coclass of the successive quotients exists and is finite. An infinite pro-p group S of coclass r is a p-adic pre-space group [23,Dfn.7.4.11,p.147], since it has a normal subgroup T , the translation group, which is a free module over the ring Z p of p-adic integers of uniquely determined rank d, the dimension, such that the quotient P = S/T is a finite p-group, the point group, which acts on T uniserially. The dimension is given by A central finiteness result for infinite pro-p groups of coclass r is provided by the so-called Theorem D, which is one of the five Coclass Theorems proved in 1994 independently by A. Shalev [27] and by C.R. 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-p groups of coclass r, for any fixed prime p and any fixed nonnegative integer r. As a consequence, if S is an infinite pro-p group of coclass r, then there exists a minimal integer i ≥ 1 such that the following three conditions are satisfied for any integer j ≥ i.
• cc(S/γ j (S)) = r, • S/γ j (S) is not a lower central quotient of any infinite pro-p group of coclass r which is not isomorphic to S, • γ j /γ j+1 (S) is cyclic of order p. The descendant tree T (R), with respect to the parent definition (P2), of the root R = S/γ i (S) with minimal i is called the coclass tree T (S) of S and its unique maximal infinite (reverse-directed) path is called the mainline (or trunk) of the tree.

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 Figures 2, resp. 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.
❄ three siblings bifurcation: If the descendant tree is a coclass tree T (R) with root R = R 0 and with mainline vertices (R n ) n≥0 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 B(R n ) with root R n , for any n ≥ 0. 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 B(2), B(4) both have depth 0, and B(5) ≃ B(7), resp. B(6) ≃ B (8), are isomorphic as trees.
If all vertices of depth bigger than a given integer k ≥ 0 are removed from branch B(n), then we obtain the (depth-)pruned branch B k (n). Correspondingly, the pruned coclass tree T k (R), resp. the entire coclass tree T (R), consists of the infinite sequence of its pruned branches (B k (n)) n≥0 , resp. branches (B(n)) n≥0 , connected by the mainline, whose vertices R n are called infinitely capable.

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 [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-p group S of coclass r ≥ 1 and dimension d, and for any given depth k ≥ 1, there exists an effective minimal lower bound f (k) ≥ 1, where periodicity of length d of depth-k pruned branches of the coclass tree T (S) sets in, that is, there exist graph isomorphisms 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 P = R f (k) is called the periodic root of the pruned coclass tree, for a fixed value of the depth k. See Figure 1.

Multifurcation and coclass graphs
Assume that parents of finite p-groups are defined as last non-trivial lower central quotients (P2). For a p-group G of coclass cc(G) = r, we can distinguish its (entire) descendant tree T (G) and its coclass-r descendant tree T r (G), the subtree consisting of descendants of coclass r only. The group G is coclass-settled if T (G) = T r (G).
The nuclear rank ν(G) of G (see § 14) in the theory of the p-group generation algorithm by M.F. Newman [8] and E.A. O'Brien [9] provides the following criteria.
• G is terminal, and thus trivially coclass-settled, if and only if ν(G) = 0.
• If ν(G) = 1, then G is capable, but it remains unknown whether G is coclass-settled.
• If ν(G) = m ≥ 2, then G is capable and certainly not coclass-settled. In the last case, a more precise assertion is possible: If G has coclass r and nuclear rank ν(G) = m ≥ 2, then it gives rise to an m-fold multifurcation into a regular coclass-r descendant tree T r (G) and m − 1 irregular descendant trees T r+j (G) of coclass r + j, for 1 ≤ j ≤ m − 1. 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, c = cl(Q) = cl(P )+ 1, from a parent P = π(Q) to any immediate descendant Q, the coclass remains stable, r = cc(Q) = cc(P ), if |γ c (Q)| = p. In this case, Q is a regular immediate descendant with directed edge P ← Q of depth 1, as usual. However, the coclass increases by m − 1, if |γ c (Q)| = p m with m ≥ 2. Then Q is called an irregular immediate descendant with directed edge of depth m.
If the condition of depth (or step size) 1 is imposed on all directed edges, then the maximal descendant tree T (1) of the trivial group 1 splits into a countably infinite disjoint union (8.2) T (1) =∪ ∞ r=0 G(p, r) of directed coclass graphs G(p, r), which are rather forests than trees. More precisely, the above mentioned Coclass Theorems imply that is the disjoint union of finitely many coclass trees T (S i ) of pairwise non-isomorphic infinite pro-p groups S i of coclass r (Theorem D) and a finite subgraph G 0 (p, r) of sporadic groups lying outside of any coclass tree.

Identifiers
The SmallGroups Library identifiers of finite groups, in particular p-groups, given in the form  [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 counting number .
Depending on the prime p, there is an upper bound on the order of groups for which a Small-Group identifier exists, e.g. 512 = 2 9 for p = 2, and 2187 = 3 7 for p = 3. For groups of bigger orders, a notation with generalized identifiers resembling the descendant structure is employed: A regular immediate descendant, connected by an edge of depth 1 with its parent P , is denoted by P − #1; counting number, and an irregular immediate descendant, connected by an edge of depth d ≥ 2 with its parent P , is denoted by P − #d; counting number. The ANUPQ package [11] containing the implementation of the p-group generation algorithm uses this notation, which goes back to J.A. Ascione in 1979 [18].

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. of finite p-groups of coclass 0 does not contain a coclass tree and consists of the trivial group 1 and the cyclic group C p of order p, which is a leaf (however, it is capable with respect to the lower exponent-p central series). For p = 2 the SmallGroup identifier of C p is 2, 1 , for p = 3 it is 3, 1 .
10.2. Coclass 1. The coclass graph of finite p-groups of coclass 1 consists of the unique coclass tree with root R = C p × C p , the elementary abelian p-group of rank 2, and a single isolated vertex (a terminal orphan without proper parent in the same coclass graph, since the directed edge to the trivial group 1 has depth 2), the cyclic group C p 2 of order p 2 in the sporadic part G 0 (p, 1) (however, this group is capable with respect to the lower exponent-p central series). The tree T 1 (R) = T 1 (S 1 ) is the coclass tree of the unique infinite pro-p group S 1 of coclass 1. For p = 2, resp. p = 3, the SmallGroup identifier of the root R is 4, 2 , resp. 9, 2 , and a tree diagram of the coclass graph from branch B(2) up to branch B(7) (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 p 3 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 G(2, 1), resp. G(3, 1), has periodic root 8, 3 and period of length 1 starting with branch B(3), resp. periodic root 81, 9 and period of length 2 starting with branch B(4). Both trees have branches of bounded depth 1, so their virtual periodicity is in fact a strict periodicity. The Γ s , resp. Φ s , denote isoclinism families [29,30].
However, the coclass tree of G(p, 1) with p ≥ 5 has unbounded depth and contains nonmetabelian groups, and the coclass tree of G(p, 1) with p ≥ 7 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 Figures  2 and 3 can be endowed with additional information and redrawn as structured descendant trees   The concrete examples G(2, 1) and G(3, 1) 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 G is generated by two elements x, y but the presentation contains the series of higher commutators s j , 2 ≤ j ≤ n − 1 = cl(G), starting with the main commutator s 2 = [y, x]. The nilpotency is formally expressed by s n = 1, when the group is of order |G| = p n .
For p = 3, there are three parameters 0 ≤ a ≤ 1 and −1 ≤ w, z ≤ 1 and the pc-presentation is given by 3-groups with parameter a = 0 possess an abelian maximal subgroup, those with parameter a = 1 do not. More precisely, an existing abelian maximal subgroup is unique, except for the two extra special groups G 3 0 (0, 0) and G 3 0 (0, 1), where all four maximal subgroups are abelian. In contrast to any bigger coclass r ≥ 2, the coclass graph G(p, 1) exclusively contains p-groups G with abelianization G/G ′ of type (p, p), except for its unique isolated vertex. The case p = 2 is distinguished by the truth of the reverse statement: Any 2-group with abelianization of type (2, 2) is of coclass 1 (O. 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 G(p, r) with r ≥ 2 is not uniform. p-groups with several distinct abelianizations contribute to its constitution. For coclass r = 2, there are essential contributions from groups G with abelianizations G/G ′ of the types (p, p), (p 2 , p), (p, p, p), and an isolated contribution by the cyclic group of order p 3 : 10.3.1. Abelianization of type (p, p). As opposed to p-groups of coclass 2 with abelianization of type (p 2 , p) or (p, p, p), which arise as regular descendants of abelian p-groups of the same types, p-groups of coclass 2 with abelianization of type (p, p) arise from irregular descendants of a nonabelian p-group with coclass 1 and nuclear rank 2.
For the prime p = 2, such groups do not exist at all, since the dihedral group 8, 3 is coclasssettled, which is the deeper reason for Taussky's Theorem. This remarkable fact has been observed by G. Bagnera [33, Part 2, § 4, p.182] in 1898 already.
For odd primes p ≥ 3, the existence of p-groups of coclass 2 with abelianization of type (p, p) is due to the fact that the extra special group G 3 0 (0, 0) is not coclass-settled. Its nuclear rank equals 2, which gives rise to a bifurcation of the descendant tree T (G 3 0 (0, 0)) into two coclass graphs. The regular component T 1 (G 3 0 (0, 0)) is a subtree of the unique tree T 1 (C p × C p ) in the coclass graph G(p, 1). The irregular component T 2 (G 3 0 (0, 0)) becomes a subgraph G = G (p,p) (p, 2) of the coclass graph G(p, 2) when the connecting edges of depth 2 of the irregular immediate descendants of G 3 0 (0, 0) are removed. For p = 3, this subgraph G is drawn in  2 187 3 7 Displaying additional information on kernels and targets of Artin transfers [3], we can draw these trees as structured descendant trees 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 r(G) = dim Fp (H 2 (G, F p )) coincides with its generator rank d(G) = dim Fp (H 1 (G, F p )). A σ-group is a pro-p group G which possesses an automorphism σ ∈ Aut(G) inducing the inversion x → x −1 on its abelianization G/G ′ . A Schur σ-group [34,35,7,36] is a Schur group G which is also a σ-group and has a finite abelianization G/G ′ .
It should be pointed out that 243, 3 is not root of a coclass tree, since its immediate descendant 729, 40 = B, which is root of a coclass tree with metabelian mainline vertices, has two siblings 729, 35 = I, resp. 729, 34 = H, 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. parameters abelianization class-2 quotient class-3 quotient class-4 quotient . Their finite quotients generate all mainline vertices with bicyclic centres of type (3, 3) of six coclass trees in the coclass graph G(3, 2). The association of parameters to the roots of these six trees is given in Table 1, the tree diagrams are indicated in Figures 4 and 5, and the parametrized pro-3 presentation is given by Here, it should be emphasized that 243, 13 is not root of a coclass tree, since aside from its descendant 2187, 319 , which is root of a coclass tree with metabelian mainline vertices, it possesses five further descendants 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)].  , resp. the pro-3 group given by the parameters (f, g, h) = (0, 0, 0) in Table 1. For p = 2, 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, p-groups with several distinct abelianizations contribute to the constitution of the coclass graph G(p, 3) . There are regular, resp. irregular, essential contributions Edges of depth 2 forming the interface between G(2, 2) and G(2, 3) Edges of depth 2 forming the interface between G(2, 3) and G(2, 4) . . .  10.4.1. Abelianization of type (p, p, p). Since the elementary abelian p-group C p × C p × C p of rank 3, that is, 8, 5 , resp. 27, 5 , for p = 2, resp. p = 3, is not coclass-settled, it gives rise to a multifurcation. The regular component T 2 (C p × C p × C p ) has been described in the section about coclass 2. The irregular component T 3 (C p × C p × C p ) becomes a subgraph G = G (p,p,p) (p, 3) of the coclass graph G(p, 3) when the connecting edges of depth 2 of the irregular immediate descendants of C p × C p × C p are removed. For p = 2, this subgraph G is contained in Figure 6. It has nine top level vertices of order 32 = 2 5 which can be divided into terminal and capable vertices: •  ( 32,29 ) is associated with family #75, T 3 ( 32, 30 ) is associated with family #76, T 3 ( 32, 31 ) is associated with family #77, 32, 34 gives rise to T 3 ( 64, 174 ) associated with family #78 (see § 21), and finally T 3 ( 32, 35 ) is associated with family #79 (see Figure 6). The roots of the coclass trees T 4 ( 128, 438 ) in Figure 6 and T 4 ( 128, 444|445 ) in Figure 7 are siblings. Table 2. Class-2 Quotients Q of certain metabelian 2-Groups G of Type (2, 2, 2)

SmallGroups
Hall Senior Schur multiplier 2-rank of   [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 G are denoted by H i , for 1 ≤ i ≤ 7.

History of descendant trees
Descendant trees with central quotients as parents (P1) are implicit in P. Hall's 1940 paper [30] about isoclinism of groups. Trees with last non-trivial lower central quotients as parents (P2) were first presented by C. R. Leedham-Green at the International Congress of Mathematicians in Vancouver, 1974 [20]. The first extensive tree diagrams have been drawn manually by J. A. Ascione, G. Havas and Leedham-Green (1977) [14], by Ascione (1979) [18] and by B. 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 construction: p-group generation algorithm
The p-group generation algorithm by M.F. Newman [8] and E.A. 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.

Lower exponent-p central series
For a finite p-group G, the lower exponent-p central series (briefly lower p-central series) of G is a descending series (P j (G)) j≥0 of characteristic subgroups of G, defined recursively by (13.1) P 0 (G) := G and P j (G) := [P j−1 (G), G] · P j−1 (G) p , for j ≥ 1.
Since any non-trivial finite p-group G > 1 is nilpotent, there exists an integer c ≥ 1 such that P c−1 (G) > P c (G) = 1 and cl p (G) := c is called the exponent-p class (briefly p-class) of G. Only the trivial group 1 has cl p (1) = 0. Generally, for any finite p-group G, its p-class can be defined as cl p (G) := min{c ≥ 0 | P c (G) = 1}. The complete lower p-central series of G is therefore given by For the convenience of the reader and for pointing out the shifted numeration, we recall that the (usual) lower central series of G is also a descending series (γ j (G)) j≥1 of characteristic subgroups of G, defined recursively by The following rules should be remembered for the exponent-p class: Let G be a finite p-group. (R1) cl(G) ≤ cl p (G), since the γ j (G) descend more quickly than the P j (G). (R2) ϑ ∈ Hom(G,G), for some groupG ⇒ ϑ(P j (G)) = P j (ϑ(G)), for any j ≥ 0. (R3) For any c ≥ 0, the conditions N G and cl p (G/N ) = c imply P c (G) ≤ N . (R4) For any c ≥ 0, cl p (G) = c ⇒ cl p (G/P k (G)) = min(k, c), for all k ≥ 0, in particular, cl p (G/P k (G)) = k, for all 0 ≤ k ≤ c.
We point out that every non-trivial finite p-group G > 1 defines a maximal path with respect to the parent definition (P3), consisting of c edges, · · · → π c−1 (G) = G/P 1 (G) → π c (G) = G/P 0 (G) = G/G = 1 and ending in the trivial group π c (G) = 1. The last but one quotient of the maximal path of G is the elementary abelian p-group π c−1 (G (H 1 (G, F p )) denotes the generator rank of G.

p-covering group, p-multiplicator and nucleus
Let G be a finite p-group with d generators. Our goal is to compile a complete list of pairwise non-isomorphic immediate descendants of G. It turned out that all immediate descendants can be obtained as quotients of a certain extension G * of G which is called the p-covering group of G and can be constructed in the following manner.
We can certainly find a presentation of G in the form of an exact sequence where F denotes the free group with d generators and ϑ : F −→ G is an epimorphism with kernel R := ker(ϑ). Then R ⊳ F is a normal subgroup of F consisting of the defining relations for G ≃ F/R. For elements r ∈ R and f ∈ F , the conjugate f −1 rf ∈ R and thus also the commutator [r, f ] = r −1 f −1 rf ∈ R are contained in R. Consequently, R * := [R, F ] · R p is a characteristic subgroup of R, and the p-multiplicator R/R * of G is an elementary abelian p-group, since of G is bounded from above by the p-multiplicator rank.
In particular, an immediate descendant H of G is a p-elementary abelian central extension A subgroup M/R * ≤ R/R * of the p-multiplicator of G is called allowable if it is given by the kernel M/R * = ker(ψ * ) of an epimorphism ψ * : G * → H onto an immediate descendant H of G. An equivalent characterization is that 1 < M/R * < R/R * is a proper subgroup which supplements the nucleus Therefore, the first part of our goal to compile a list of all immediate descendants of G is done, when we have constructed all allowable subgroups of R/R * which supplement the nucleus P c (G * ) = P c (F ) · R * /R * , where c = cl p (G). However, in general the list where G * /(M/R * ) = (F/R * )/(M/R * ) ≃ F/M will be redundant, due to isomorphisms F/M 1 ≃ F/M 2 among the immediate descendants.

Orbits under extended automorphisms
Two allowable subgroups M 1 /R * and M 2 /R * are called equivalent if the quotients F/M 1 ≃ F/M 2 , that are the corresponding immediate descendants of G, are isomorphic.
Such an isomorphism ϕ : F/M 1 → F/M 2 between immediate descendants of G = F/R with c = cl p (G) has the property that ϕ(R/M 1 ) = ϕ(P c (F/M 1 )) = P c (ϕ(F/M 1 )) = P c (F/M 2 ) = R/M 2 and thus induces an automorphism α ∈ Aut(G) of G which can be extended to an automorphism α * ∈ Aut(G * ) of the p-covering group G * = F/R * of G. The restriction of this extended automorphism α * to the p-multiplicator R/R * of G is determined uniquely by α. Since α * (M/R * ) · P c (F/R * ) = α * [M/R * · P c (F/R * )] = α * (R/R * ) = R/R * , according to the rule (R2), each extended automorphism α * ∈ Aut(G * ) induces a permutationα of the allowable subgroups M/R * ≤ R/R * . We define (16.1) P := α | α ∈ Aut(G) to be the permutation group generated by all permutations induced by automorphisms of G. Then the map Aut(G) → P , α →α is an epimorphism and the equivalence classes of allowable subgroups M/R * ≤ R/R * are precisely the orbits of allowable subgroups under the action of the permutation group P . Eventually, our goal to compile a list {F/M i | 1 ≤ i ≤ N } of all immediate descendants of G will be done, when we select a representative M i /R * for each of the N orbits of allowable subgroups of R/R * under the action of P . This is precisely what the p-group generation algorithm does in a single step of the recursive procedure for constructing the descendant tree of an assigned root.

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 ν(G) of G admits a decision about the capability of G: • For the related phenomenon of multifurcation of a descendant tree at a vertex G with nuclear rank ν(G) ≥ 2 see § 8 on multifurcation and coclass graphs. The p-group generation algorithm provides the flexibility to restrict the construction of immediate descendants to those of a single fixed step size 1 ≤ s ≤ ν, which is very convenient in the case of huge descendant numbers (see the next section).

Numbers of immediate descendants
We denote the number of all immediate descendants, resp. immediate descendants of step size s, of G by N , resp. N s . Then we have N = ν s=1 N s . As concrete examples, we present some interesting finite metabelian p-groups with extensive sets of immediate descendants, using the SmallGroups identifiers and additionally pointing out the numbers 0 ≤ C s ≤ N s of capable immediate descendants in the usual format as given by actual implementations of the p-group generation algorithm in the computational algebra systems GAP and MAGMA. These invariants completely determine the local structure of the descendant tree T (G). First, let p = 2. We begin with groups having abelianization of type (2, 2, 2). See Figure 6.

Schur multiplier
Via the isomorphism the quotient 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 G = F/R as in the previous section. Then the second cohomology group I.R. Shafarevich [38, § 6, p.146] has proved that the difference between the relation rank r(G) = dim Fp (H 2 (G, F p )) of G and the generator rank d(G) = dim Fp (H 1 (G, F p )) of G is given by the minimal number of generators of the Schur multiplier of G, that is for all quotients G j := G/P j (G) of p-class cl p (G j ) = j, j ≥ 0, of a pro-p group G with finite abelianization G/G ′ . Furthermore, J. Blackhurst (in the appendix On the nucleus of certain p-groups of a paper by N. Boston, M.R. Bush and F. Hajir [35]) has proved that a non-cyclic finite p-group G with trivial Schur multiplier M (G) is a terminal vertex in the descendant tree T (1) of the trivial group 1, that is,

Pruning strategies
For searching a particular group in a descendant tree T (R) 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 (F1) filtering the σ-groups (see Definition 10.1), (F2) eliminating a set of certain transfer kernel types (TKTs, see [6, pp.403-404]), (F3) cancelling all non-metabelian groups (thus restricting to the metabelian skeleton), (F4) removing metabelian groups with cyclic centre (usually of higher complexity), (F5) cutting off vertices whose distance from the mainline (depth) exceeds some lower bound, (F6) combining several different sifting criteria.
The result of such a sieving procedure is called a pruned descendant tree T * (R) < T (R) 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.

Striking news: periodic bifurcations in trees
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 G/G ′ ≃ (2, 2, 2). The 2-groups under investigation are threegenerator groups with elementary abelian commutator factor group of type (2, 2, 2). As shown in Figure 6 of § 10, all such groups are descendants of the abelian root 8, 5 . Among its immediate descendants of step size 2, there are three groups which reveal multifurcation. 32, 27 has nuclear rank ν = 3, giving rise to 3-fold multifurcation. The two groups 32, 28 and 32, 34 possess the required nuclear rank ν = 2 for bifurcation. Due to the arithmetical origin of the problem, we focused on the latter, G := 32, 34 , and constructed an extensive finite part of its pruned descendant tree T * (G), using the p-group generation algorithm [8,9,10] as implemented in the computational algebra system Magma [43,44,13]. All groups turned out to be metabelian.
Remark 21.1. Since our primary intention is to provide a sound group theoretic background for several phenomena 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 T (G) to the pruned descendant tree T * (G) filters all vertices which satisfy one of the conditions in Equations (21.3) or (21.8), and essentially consists of pruning strategy (F2), more precisely, of (1) omitting all the 13 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 x, y, z the generators of a finite 2-group G = x, y, z with abelian type invariants (2, 2, 2). We fix an ordering of the seven maximal normal subgroups by putting (21.1) Just within this subsection, we select a special designation for a TKT [6, pp.403-404] whose first layer consists exactly of all these seven planes in the 3-dimensional F 2 -vector space G/G ′ , in any ordering.
In view of forthcoming number theoretic applications, we add the following Corollary 21.1. Let 0 ≤ j ≤ ℓ be a non-negative integer.
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 T * (δ j (G)) are all isomorphic to T * (G) as graphs. This is visualized impressively by Figure 7.
The extent to which we constructed the pruned descendant tree suggests the following conjecture. One-parameter polycyclic pc-presentations for all occurring groups are given as follows.
To obtain a presentation for the vertices δ r−3 (G)(−#1; 1) c−r − #1; 2, c ≥ r, at depth 1 in the distinguished periodic sequence whose vertices are characterized by the permutation TKT (21.8), we must only add the single relation x 2 = s c to the presentation (21.13) of the mainline vertices of the coclass tree T r (δ r−3 (G)) given in Theorem 21.2.
These 3-groups are two-generator groups of coclass at least 2 with elementary abelian commutator quotient of type (3,3). As shown in Figure 4 of § 10, all such groups are descendants of the extra special group 27, 3 . 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 243, 6 and 243, 8 do not show multifurcation themselves but they are not coclass-settled either, since their immediate mainline descendants Q = 729, 49 and U = 729, 54 possess the required nuclear rank ν = 2 for bifurcation. We constructed an extensive finite part of their pruned descendant trees T * (G), G ∈ {Q, U }, using the p-group generation algorithm [8,9,10] as implemented in the computational algebra system Magma [43,44,13].
Denote by x, y the generators of a finite 3-group G = x, y with abelian type invariants (3,3). We fix an ordering of the four maximal normal subgroups by putting (21.14) H Within this subsection, we make use of special designations for transfer kernel types (TKTs) which were defined generally in [6, pp.403-404] and more specifically for the present scenario in [50,4].
We point out that, for instance E. Remark 21.3. We choose the following sifting strategy for reducing the entire descendant tree T (G) to the pruned descendant tree T * (G). We filter all vertices which, firstly, are σ-groups, and secondly satisfy one of the conditions in Equations (21.17) or (21.26), 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 σgroups, and omitting 2 of the 3 capable step size-2 descendants having TKT H.4, resp. G.16, together with their complete descendant trees, and (2) eliminating 2 of the 5 terminal step size-1 descendants having TKT c.18, resp. c.21, and 2 of the 3 capable step size-1 descendants having TKT H.4, resp. G.16, in Theorem 21.3.
Theorem 21.3 provided the scaffold of the pruned descendant tree T * (G) of G = 243, n , for n ∈ {6, 8}, 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.
The metabelian 3-groups forming the three distinguished periodic sequences V 0,2k = δ 1 (G)(−#1; 1) 2k − #1; 2|4|6 resp. 4|5|6 with k ≥ 0 of the pruned coclass tree T 2 * (G) in Corollary 21.2, for i = 0, 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 T (G) \ T 2 (G) are of derived length 3, the set of these descendants can be defined in the following way.
Definition 21.2. Let P be a finite p-group, then the set of all finite p-groups D whose second derived quotient D/D ′′ is isomorphic to P is called the cover cov(P ) of P . The subset cov * (P ) consisting of all Schur σ-groups in cov(P ) is called the balanced cover of P . Corollary 21.3. For 0 ≤ k ≤ ℓ, the group V 0,2k , which does not have a balanced presentation, possesses a finite cover of cardinality #cov(V 0,2k ) = k + 1 and a unique Schur σ-group in its balanced cover #cov * (V 0,2k ) = 1. More precisely, the covers are given explicitly by (21.27) cov The arrows in Figures 8 and 9 indicate the projections π from all members D of a cover cov(P ) onto the common metabelianization P , that is, in the sense of the parent definition (P4), from the descendants D onto the parent P = π(D). The p-group generation algorithm [8,9,10], which is implemented in the computational algebra system Magma [43,44,13], was used for constructing the pruned descendant trees T * (G) with roots G = 243, 6|8 which were defined as the disjoint union of all pruned coclass trees T j+2 * (δ 2j+1 (G)) of the descendants δ 2j+1 (G) = G(−#1; 1−#2; 1) j −#1; 1, 0 ≤ j ≤ 10, of G as roots, together with 4 siblings in the irregular component T j+3 * (δ 2j+1 (G)), 3 of them Schur σ-groups with µ = 2 and ν = 0. Using the strict periodicity [1, 2] of each pruned coclass tree T j+2 * (δ 2j+1 (G)), 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.

Conclusion
We emphasize that the results of section 21.2 provide the background for considerably stronger assertions than those made in [7] (which were, however, sufficient already to disprove erroneous claims in [48,49]). Firstly, since they concern four TKTs E.6, E.14, E.8, E.9 instead of just TKT E.9, and secondly, since they apply to varying odd nilpotency class 5 ≤ cl(G) ≤ 19 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.