Co-periodicity isomorphisms between forests of finite p-groups

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 quotient, the information content of each coclass subtree with metabelian mainline 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.


Introduction
Denote by T the rooted tree of all finite 3-groups G with elementary bicyclic commutator quotient G/G ′ ≃ C 3 × C 3 , and let T * be the infinite pruned subtree of T , where all descendants of capable non-metabelian vertices are eliminated. The main intention of this paper is to prove that the information content of the tree T * can be reduced to a finite set of representatives with the aid of two kinds of periodicity.
• Firstly, the well-known virtual periodicity isomorphisms B(n + ℓ) ≃ B(n) between the finite depth-pruned branches B(n), n ≥ n * , of a coclass subtree T r ⊂ T * are refined to strict periodicity isomorphisms between complete branches which reduce the information content of the infinite coclass subtree to the finite union of pre-period (B(n)) n * ≤n<p * and first primitive period (B(n)) p * ≤n<p * +ℓ . The virtual periodicity was proved by du Sautoy [1] and independently by Eick and Leedham-Green [2] for groups of any prime power order. The strict periodicity for p = 3 and type C 3 × C 3 is proved in the present paper. • Secondly, evidence is provided of co-periodicity isomorphisms F (r +2) ≃ F (r) between the infinite coclass forests F (r), r ≥ 1, which reduce the information content of the pruned tree T * to the union of pre-period (F (r)) 1≤r≤4 and first primitive period (F (r)) 5≤r≤6 , consisting of the leading six coclass forests only. The discovery of this co-periodicity is the progressive innovation in the present paper. Together with the coclass theorems of Leedham-Green [3] and Shalev [4], which imply that each coclass forest F (r) consists of a finite sporadic part F 0 (r) and a finite number of coclass trees T r j , 1 ≤ j ≤ t, each having a finite information content due to the strict periodicity, this shows that the pruned infinite subtree T * of the tree T is described by finitely many representatives only.
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 [1,2,5] and the recently discovered co-periodicity isomorphisms between infinite coclass forests in § 3. The behavior of algebraic invariants of the tree vertices and their automorphism groups is described with simple transformation laws in § 4. The graph theoretic preliminaries are supplemented by connections between depth, width, information content and numbers of immediate descendants in § 5, identifiers of groups in § 6, and precise definitions of mainlines and sporadic parts in § 7. The main theorems are presented in § 8.
Then we focus on the tree T of finite 3-groups G with abelianization G/G ′ ≃ C 3 × C 3 . The flow of our investigations is guided by § 10 concerning the remarkable infinite main trunk (P 2r−1 ) r≥2 of certain metabelian vertices in T which gives rise to the top vertices of all coclass forests F (r), r ≥ 2, by periodic bifurcations and constitutes the germ of the newly discovered co-periodicity F (r + 2) ≃ F (r) of length two. To start with a beautiful highlight, we immediately celebrate the simple structure of the first primitive period (F (r)) 5≤r≤6 in § § 11 and 12 and defer the somewhat arduous task of describing the exceptional pre-period (F (r)) 1≤r≤4 to the concluding § § 13 and 14.
Finally, we point out that our theory, together with the investigations of Eick [6], provides an independent verification and confirmation of all results about the metabelian skeleton M of the tree T in the dissertation of Nebelung [7], since M is a subtree of the pruned tree T * . The present paper shows the co-periodicity of the sporadic parts F 0 (r) and coclass trees T r j , 1 ≤ j ≤ t, of the coclass forests F (r), and [6, § 5.2, pp. 114-116] establishes the connection between the coclass trees T r j and infinite metabelian pro-3 groups of coclass r.

Descendant trees and coclass forests
Let p be a prime number. In the mathematical theory of finite groups of order a power of p, socalled p-groups, the introduction of the parent-child relation by Leedham-Green and Newman [8, pp. 194-195] has simplified the classification of such groups considerably. The relation is defined in terms of the lower central series (γ i G) i≥1 of a p-group G, where is called the parent of G, and G is a child (or immediate descendant ) of πG.

Definition 2.2.
For an assigned finite p-group R > 1, the descendant tree T (R) with root R is defined as the digraph (V, E) whose set of vertices V consists of all isomorphism classes of p-groups G with G/γ i G ≃ R, for some 2 ≤ i ≤ cl(G) + 1, and whose set of directed edges E consists of all child-parent pairs (2.6) (G → πG) := (G, πG) ∈ V × V.
The mapping π : V \ {R} → V is called the parent operator.
If the root R is abelian, then all vertices of the tree T (R) share the common abelianization G/G ′ ≃ R. Since a nilpotent group with cyclic abelianization is abelian, the descendant tree T (R) of a cyclic root R > 1 consists of the single isolated vertex R. The classification of p-groups by their abelianization is refined further, if directed edges are restricted to starting vertices G with cyclic last non-trivial lower central γ c G of order p. Then the descendant tree of R splits into a countably infinite disjoint union of directed subgraphs, where ord(R) = p e and the vertices of the component G(p, r, R) with fixed r ≥ e − 1 share the same coclass, cc = r, as a common invariant, since the logarithmic order lo := (log p • ord) and the nilpotency class cl of the parent πG and child G satisfy the rule According to the coclass theorems by Leedham-Green [3] and Shalev [4], a coclass graph G(p, r, R) is the disjoint union of a finite sporadic part G 0 (p, r, R) and finitely many coclass trees T r j (with infinite mainlines), that is, a forest for which there exist integerss,t ≥ 0 such that (2.9) G(p, r, R) = G 0 (p, r, R)∪ ˙ t j=1 T r j with #G 0 (p, r, R) =s.
Definition 2.4. In the present paper, the focus will lie on finite p-groups with fixed prime p = 3 arising as descendants of the fixed elementary bicyclic 3-group R := C 3 ×C 3 of order 3 e with e = 2, where C n denotes the cyclic group of order n. This assumption permits a simplified notation by omitting the explicit mention of p and R. Further, we shall slightly reduce the complexity of the forests G(r) := G(3, r, C 3 × C 3 ), r ≥ e − 1 = 1, by eliminating the descendants of capable (i.e., non-terminal) non-metabelian vertices. This pruned light-weight version of G(r) will be denoted by F (r), called the coclass-r forest, and Formula (2.9) becomes (2.10) T r j with #F 0 (r) = s and possibly different integers s =s and 1 ≤ t ≤t.
Remark 2.1. In § § 11 and 12 it will turn out that the coclass trees T r j with metabelian mainlines do not contain any capable non-metabelian vertices. So the pruning process from G(r) to F (r) concerns the sporadic part F 0 (r), and reduces the number t ≤t of coclass trees by eliminating those with non-metabelian mainlines entirely, but does not affect the coclass trees with metabelian mainlines, which remain complete in spite of pruning.
When G = (V, E) is a finite digraph with vertex cardinality #(V ) = n ∈ N, we can identify V with the set {1, . . . , n}. Then the set of directed edges E ⊂ V × V is characterized uniquely by the characteristic function χ E of E in V × V = {1, . . . , n} × {1, . . . , n}, which is called the n × n adjacency matrix A = (a i,j ) 1≤i,j≤n of G. Its entries are defined, for all 1 ≤ i, j ≤ n, by Proposition 3.1. Let G = (V, E) andG = (Ṽ ,Ẽ) be two finite digraphs with n vertices. Then G andG are isomorphic if and only if there exists a bijection ψ : V →Ṽ such that the entries of the adjacency matrices a i,j = χ E (i, j) = χẼ(ψ(i), ψ(j)) =ã ψ(i),ψ(j) coincide for all 1 ≤ i, j ≤ n.
The in-resp. out-degree of a vertex v ∈ V in a finite digraph can be expressed in terms of the vth column-resp. row-sum of the adjacency matrix: In particular, if G = T (R) is a finite directed in-tree with root R, then each row of the adjacency matrix A corresponding to a vertex v = R contains a unique 1 and Proposition 3.2. Let T (R) = (V, E) andT (R) = (Ṽ ,Ẽ) be two rooted directed in-trees, and denote by π andπ their parent operators. Then a bijection ψ : V →Ṽ with ψ(R) =R is an isomorphism of rooted directed in-trees if and only if ψ(π(v)) =π(ψ(v)) for all v ∈ V \ {R}, that is, ψ • π =π • ψ (briefly: ψ commutes with the parent operator), as shown in Figure 1.

Algebraically structured digraphs
4.1. General invariants and their transformation laws. 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 φ, the transformation law, when a graph isomorphism is applied. This behaviour is made precise in the following definitions.
Definition 4.1. Let G = (V, E) be a graph. Suppose that X = ∅ is a set, and each vertex v ∈ V is associated with some kind of information S(v) ∈ X. Then (G, S) is called a structured graph with respect to the mapping S : V → X, v → S(v).
In particular, if the setsX = X coincide and φ = 1 X is the identity mapping of the set X, then ψ is called a strict isomorphism of structured digraphs, and it satisfies the relationS • ψ = S. Let G = (V, E) andG = (Ṽ ,Ẽ) be two structured digraphs with structure mappings S : V → X andS :Ṽ →X, and let ψ : V →Ṽ be a φ-isomorphism of the two structured digraphs with respect to a mapping φ : X →X, that is,S • ψ = φ • S. Then S is called a φ-invariant under ψ (or invariant under the isomorphism ψ and transformation law φ). In particular, ifX = X, φ = 1 X andS • ψ = S, then S is called a strict invariant under ψ.

4.2.
Algebraic invariants considered in this paper. With respect to applications in other mathematical theories, in particular, algebraic number theory and class field theory, certain properties of the automorphism group Aut(G) of a finite 3-group G are crucial. The general frame of these aspects is the following. 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 σ-group, if there exists a generator inverting automorphism σ ∈ Aut(G) such that σ(x) ≡ x −1 mod G ′ , for all x ∈ G, or equivalently σ(x) = x −1 , for all x ∈ H 1 (G, F p ). If additionally σ(x) = x −1 , for all x ∈ H 2 (G, F p ), then G is called a group with RI-action or group with relator inverting automorphism. If Aut(G) contains a bicyclic subgroup C 2 × C 2 , then we call G a group with V 4 -action. It is convenient to define the action flag of G by 2 if G possesses GI-action and V 4 -action, starred (2 * ) for RI-action, 1 if G possesses GI-action but no V 4 -action, starred (1 * ) for RI-action, 0 if G has no GI-action and no V 4 -action.
Remark 4.1. Suppose that G is a finite p-group with odd prime p. We point out that 2 divides the order #Aut(G), if G is a group with GI-action, but the converse claim may be false. If G is a group with V 4 -action, then 4 divides #Aut(G), but we emphasize that the converse statement, even in the case that 8 divides #Aut(G), may be false, when Aut(G) contains a cyclic group C 4 or a (generalized) quaternion group Q(2 e ) of order 2 e with e ≥ 3.
For a brief description of abelian quotient invariants in logarithmic form, we need the concept of nearly homocyclic p-groups. With an arbitrary prime p ≥ 2 these groups appear in [9,p. 68,Thm. 3.4] and they are treated systematically in [7, § 2.4]. For our purpose, it suffices to consider the special case p = 3.
Definition 4.4. By the nearly homocyclic abelian 3-group A(3, n) of order 3 n , for an integer n ≥ 2, we understand the abelian group with logarithmic type invariants (q + r, q), where n = 2q + r with integers q ≥ 1 and 0 ≤ r < 2, by Euclidean division with remainder. Additionally, including two degenerate cases, we define that A(3, 1) denotes the cyclic group C 3 of order 3, and A(3, 0) denotes the trivial group 1.
The following invariants S : V → X of finite 3-groups v ∈ V with abelianization v/v ′ ≃ C 3 ×C 3 will be of particular interest in the whole paper: • The logarithmic order lo : V → N 0 , v → n := log p (ord(v)), • the nilpotency class cl : V → N 0 , connected with the index of nilpotency m by the relation • the coclass cc : • the order of the automorphism group #Aut : • the action flag σ : V → N 0 , defined by Formula (4.1), • the transfer kernel type (TKT) κ : • the abelian quotient invariants of the commutator subgroup τ 2 : Abelian quotient invariants are given in logarithmic notation. The transfer kernel type κ(v) is simplified by a family of non-negative integers, in the following way: for 1 ≤ i ≤ 4, 5. The graph theoretic structure of a tree 5.1. Cardinality of branches and layers, depth and width of a tree. The graph theoretic structure of a coclass tree T with unique infinite mainline and finite branches, consisting of isomorphism classes of finite p-groups, is described by the following concepts. The width of the tree is the maximal cardinality of its layers, Each vertex v of the branch B(e) is connected with the mainline by a unique finite path of directed edges from v to the branch root m e , formed by the iterated parents π i v of v, The length d ≥ 0 of this path is called the depth dp(v) of v. The depth of a branch B ⊂ T is the maximal depth of its vertices, Definition 5.2. Let T be a coclass tree. The depth of the tree is the maximal depth of its branches, (5.5) dp(T ) := sup{dp(B(n)) | n ≥ n * }.
Throughout this paper, we assume that both, the depth dp(T ) and the width wd(T ) of the tree, are bounded. This assumption is satisfied by all trees of finite 3-groups under investigation in the sequel. However, we point out that that tree T 1 (C 5 × C 5 ) of finite 5-groups with coclass one has unbounded depth, and the tree T 1 (C 7 × C 7 ) of finite 7-groups with coclass one even has unbounded width and depth. (Compare [6, § 5.1, pp. 113-114].) Lemma 5.1. Let e ≥ n * , B := B(e) and d := dp(B). Then Proof. Since m e is the root of the branch B(e), we have Lyr e B = {m e }, but (∀ n < e) Lyr n B = ∅.
Since d = dp(B) = max{dp(v) | v ∈ B}, there exists a vertex t ∈ B, necessarily terminal if d > 0, such that dp(t) = d. The iterated parents π i t of t form the unique finite path from t to the branch root m e (see Figure 3), t = π 0 t → π 1 t → π 2 t → . . . → π d t = m e , and we have (∀ e ≤ n ≤ e + d) Lyr n B ⊇ {π e+d−n t} = ∅ but (∀ n > e + d) Lyr n B = ∅.
Lemma 5.2. Let n ≥ n * and d := dp(T ). Then Proof. Since dp(T ) = sup{dp(B(n)) | n ≥ n * }, we have dp(B(i)) ≤ dp(T ) = d for each i ≥ n * . A branch B(i) with i > n cannot contribute to Lyr n T . On the other hand, if n−d > n * , then a branch Lyr j B(i), according to Lemma 5.1, and we obtain i + dp(B(i)) ≤ i + d < n (see Figure 3). Consequently, Lyr n B(i).  Since the implementation of the p-group generation algorithm [12,13,14] in the computational algebra system MAGMA [15,16,17] is able to give the number of all, respectively only the capable, immediate descendants (children) of an assigned finite p-group, we express the cardinalities of the branches of a coclass tree, which were given in a preliminary form in Lemma 5.1, in terms of these numbers N 1 , respectively C 1 .
Theorem 5.1. Let T = (V, E) be a coclass tree with tree root R of logarithmic order lo(R) = n * , pre-period of length ℓ * ≥ 0, and period of primitive length ℓ ≥ 1. For each vertex v ∈ V , denote by N 1 (v) the number of all children (of step size s = 1) and by C 1 (v) the number of capable children of v. When m e is the vertex with lo(m e ) = e ≥ n * on the mainline of T , (1) If the tree is of depth dp(T ) = 1, then (2) If the tree is of depth dp(T ) = 2, then (3) If the tree is of depth dp(T ) = 3, then Proof. Put B := B(e). Generally, we have B = Lyr e B∪ . . .∪ Lyr e+d B with d := dp(B), according to Lemma 5.1. If dp(T ) = 1, then d ≤ 1 and B = Lyr e B∪ Lyr e+1 B. We have Lyr e B = {m e } and #Lyr e+1 B = N 1 (m e ) − 1, since the next mainline vertex m e+1 is one of the N 1 (m e ) children of m e but does not belong to B. Thus, we obtain #B = #Lyr e B + #Lyr e+1 B = 1 + N 1 (m e ) − 1 = N 1 (m e ).
For assigned small values of the depth d ≤ 3, the width can be expressed in terms of descendant numbers in the following manner: (1) If the tree is of depth d = 1, then (5.13) wd(T ) = max{N 1 (m n−1 ) | n * + 1 ≤ n ≤ n * + ℓ * + ℓ}.
(2) If the tree is of depth d = 2, then wd(T ) is the maximum among the number N 1 (m n * ) and all expressions (5.14) where n runs from n * + 2 to n * + ℓ * + ℓ + 1.
Proof. According to [1,2,5], the periodicity of the branches of a coclass tree T with root R := m n * and bounded depth d := dp(T ) and width wd(T ) can be expressed by means of isomorphisms between branches, starting from the periodic root P := m n * +ℓ * : where ℓ * ≥ 0 denotes the length of the pre-period and ℓ ≥ 1 is the primitive period length. With Lemma 5.1, an immediate consequence is the periodicity of branch layer cardinalities: According to Lemma 5.2, we have #Lyr n T = n k=max(n * ,n−d) #Lyr n B(k), and thus For finding the maximal layer cardinality, the root term #Lyr n * T = #Lyr n * B(n * ) = 1 can be omitted, since each layer contains a mainline vertex. Beginning with n = n * + d, the expression for the tree layer cardinality #Lyr n T is a sum of d + 1 terms and we must find the logarithmic order n = n * +x where periodicity of all terms sets in. This leads to the inequality n * +x−d ≥ n * +ℓ * +ℓ with solution x ≥ ℓ * + ℓ + d. Consequently, m = n * + ℓ * + ℓ + d − 1 is the biggest logarithmic order for which a new value of the tree layer cardinality #Lyr m T may occur (see Figure 3). At the logarithmic order m + 1, periodic repetitions of the values of tree layer cardinalities begin. In the special case of d ≤ 3, Theorem 5.1 yields an expression in terms of descendant numbers: #Lyr n T =#Lyr n B(n − d) + . . . + #Lyr n B(n) The following concept provides a quantitative measure for the finite information content of an infinite tree with periodic branches. where p * = n * + ℓ * denotes the logarithmic order of the periodic root P of T (see Figure 3).

Identifiers of the SmallGroups Library
Independently of being metabelian or non-metabelian, a finite 3-group G of order up to 3 8 = 6561 will be characterized by its absolute identifier G ≃ |G|, i , according to the SmallGroups Database [18,19]. Starting with order 3 9 = 19683, a group G is characterized by the absolute identifier of the parent π(G) ≃ |π(G)|, i in the SmallGroups Database [19] together with a relative identifier −#s; j generated by the ANUPQ package [20] of MAGMA [17]. Here, s denotes the step size of the directed edge G → π(G). Occasionally, certain groups of order 3 6 = 729 and coclass 2 are identified by single capital letters A, . . . , X similarly as in [21,22,23].

Mainlines of coclass trees and sporadic parts of coclass forests
If we define a mainline as a maximal path of infinitely many directed edges of step size s = 1, then there arises the ambiguity that a vertex could be root of several coclass trees. The metabelian 3-group 243, 3 = P 5 , for instance, would be the end vertex of more then one mainline, namely on the one hand of the metabelian mainline Therefore, an additional condition is required in the precise definition of a mainline. Example 7.1. The metabelian 3-group 729, 40 = B is root of the coclass-2 tree T 2 B with metabelian mainline. The metabelian 3-group 729, 35 = I is root of the coclass-2 tree T 2 I with non-metabelian mainline. According to our pruning convention that descendants of capable non-metabelian vertices do not belong to the coclass forests F (r), this tree is not an object of examination in the present paper.
Finally, the metabelian 3-group 729, 34 = H is not root of a coclass tree.
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 F (r) with r ≥ 1 is the complement of the union of the (finitely many) coclass trees in the 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 s ≥ 2 end in vertices (mainline or of depth dp ≥ 1) of a coclass tree, due to the phenomenon of multifurcation.
Example 7.2. In the second mainline vertex m 6 = 729, 49 = Q of the coclass-2 tree T 2 R 2 2 with root R 2 2 = 243, 6 , a bifurcation occurs, due to the nuclear rank ν(Q) = 2. In fact, the directed edge of step size s = 2 which ends in the vertex m 6 = Q is the final edge of an infinite path with alterating step sizes s = 2 and s = 1, due to periodic bifurcations. However, this non-metabelian path is not the topic of investigations in the present paper. For detailed information on these matters see [24] and [25].
The same is true for the second mainline vertex 729, 54 = U of the coclass-2 tree T 2 R 2 3 with root R 2 3 = 243, 8 . The unnecessary requirement of coclass stability would eliminate the pre-periods of the trees T 2 R 2 and T 2 R 3 and enforce purely periodic subtrees with periodic coclass-settled roots, namely T 2 2187, 285 ⊂ T 2 R 2 and T 2 2187, 303 ⊂ T 2 R 3 .

Two main theorems on periodicity and co-periodicity isomorphisms
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 [1,2,5] that an infinite coclass tree T r of finite p-groups with fixed coclass r ≥ 1 is the disjoint union of its branches T r =˙ ∞ n=n * B(n), which can be partitioned into a single finite pre-period˙ p * −1 n=n * B(n) of length ℓ * = p * − n * ≥ 0 and infinitely many copies of a finite primitive period˙ p * +ℓ−1 n=p * B(n) of length ℓ ≥ 1, where the integer p * ≥ n * characterizes the position of the periodic root on the mainline, provided the tree is suitably depth-pruned.
The following first main result of this paper establishes the details of the primitive period of branches of five coclass-4 trees T 4 R 4 i =˙ ∞ n=n * B(n), with n * = 9, respectively of three coclass-5 trees T 5 R 5 j =˙ ∞ n=n * B(n), with n * = 11, of finite 3-groups with mainline vertices having a single total transfer kernel and roots R 4 i := 2187, 64 − #2; n(i) with (n(i)) 2≤i≤6 = (39, 44, 54, 57, 59), respectively R 5 j := 2187, 64 − #2; 33 − #2; n(j) with (n(j)) 2≤j≤4 = (29, 37, 39), written in the notation of [18,19,20]. In fact, we prove more than the virtual periodicity for arbitrary finite pgroups in [1,2,5], since all trees of the particular finite 3-groups in our investigation have bounded depth and therefore reveal strict periodicity. For each integer n ≥ n * , there exists a bijective mapping ψ : B(n) → B(n + 2) which is a strict isomorphism of finite structured in-trees for the strict invariants in-degree in(), out-degree out(), coclass r = cc(), relation rank µ(), nuclear rank ν(), action flag σ(), and transfer kernel type κ(). Moreover, ψ is a φ-isomorphism of finite structured in-trees for the following φ-invariants with their transformation laws φ: • logarithmic order n = lo() with φ(n) = n + 2, Proof. The φ-isomorphisms between the finite branches of a tree describe the first periodicity and reduce an infinite tree to its finite primitive period, provided the periodicity is pure. This will be proved for even coclass r ≥ 4 in Theorem 11.3 for i = 39, in Thm. 11.4 for i = 44, in Thm. 11.5 for i = 54, in Thm. 11.6 for i = 57, and in Thm. 11.7 for i = 59. For odd coclass r ≥ 5, it will be proved in Theorem 12.3 for j = 29, in Thm. 12.4 for j = 37, and in Thm. 12.5 for j = 39.
Remark 8.1. According to Theorem 8.1, the diagrams of coclass-r trees (r ≥ 4) whose mainline vertices V possess a single total kernel ker(T 1 ) = V among the transfers T i : V → U i /U ′ i to the four maximal subgroups U i < V (1 ≤ i ≤ 4) reveal several surprising features: firstly, the branches are purely periodic of primitive length at most 2 without pre-period, secondly, the branches are of uniform depth 2 only, and finally, none of the vertices gives rise to descendants of coclass bigger than r. So the trees are entirely regular and coclass-stable, in contrast to the trees with 3-groups G of coclass r = cc(G) ∈ {1, 2, 3} as vertices.
Unfortunately it is much less well known that the entire metabelian skeleton M(R) of the descendant tree T (R) of the elementary bicyclic 3-group R : consists of a finite sporadic part M r 0 and finitely many metabelian coclass treesT r i , and there is a periodicity M r ≃ M r+2 for each r ≥ 3. This was proved by Nebelung [7] and confirmed by Eick [6, Cnj. 14, p. 115].
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 R Let the integer u := 19 be an upper bound. For each integer 4 ≤ r ≤ u, and for each of the six roots R r i , 1 ≤ i ≤ 6, with even coclass r ≥ 4, respectively the four roots R r j , 1 ≤ j ≤ 4, with odd coclass r ≥ 5, there exists a bijective mapping ψ : , which is a strict isomorphism of infinite structured in-trees for the strict invariants in-degree in(), out-degree out(), relation rank µ(), nuclear rank ν(), action flag σ(), and transfer kernel type κ(). Moreover, ψ is a φ-isomorphism of infinite structured in-trees for the following φ-invariants with their transformation laws φ: Proof. The statement for the metabelian skeletonsT r R r i of the coclass trees T r R r i is one of the main results of Nebelung's thesis [7]. With the aid of Theorem 8.1, the periodicity of the entire coclass trees T r R r i with 4 ≤ r ≤ 21 and fixed subscript i has been verified by computing the metabelian and non-metabelian vertices of the first four branches B r (n) with 2r + 1 ≤ n ≤ 2r + 4 of the trees T r R r i . The computations were executed by running our own program scripts for the Computer Algebra System MAGMA [17], which contains an implementation of the pgroup generation algorithm by Newman [12,26] and O'Brien [13,14], the SmallGroups Database [18,19], and the ANUPQ package [20]. It turned out that, firstly, B r (2r + 1) ≃ B r (2r + 3) and B r (2r + 2) ≃ B r (2r + 4), for each 4 ≤ r ≤ 21, and secondly, B r (2r + 1) ≃ B r+2 (2(r + 2) + 1) and The established φ-isomorphisms between the infinite coclass trees T r R r i and T r+2 R r+2 i , for 4 ≤ r ≤ 19, describe the germ of the second periodicity expressed in Conjecture 8.1. Invariants connected with the nilpotency class or coclass are not strict and are subject to the following mappings: the shifts φ(c) = c + 2 for c = cl(), φ(r) = r + 2 for r = cc(), and φ(n) = n + 4 for n = lo(), and the corresponding transformations For a = #Aut(), the transformation law is described by the homothety φ(a) = a · 3 8 .
Thus, the confidence in the validity of the following conjecture is supported extensively by sound numerical data. Consequently, all coclass trees T r R r i with r ≥ 4 and fixed subscript i are co-periodic in the variable coclass parameter r with primitive length ℓ = 2. The eight coclass trees T r R r i with r ∈ {1, 2, 3}, and i = 1 for r = 1, i ∈ {1, 2, 3} for r = 2, i ∈ {1, 2, 3, 4} for r = 3, can be viewed as the pre-period of the co-periodicity. (Compare [6, Cnj. 14, p. 115].)

Parametrized polycyclic power-commutator presentations
The general graph theoretic and algebraic foundations of the coclass forests F (r) with r ≥ 1 have been developed completely in the preceding sections 2 -7. Now we can turn to the main goal of the present paper, that is, the proof of the main theorems in section 8 by the systematic investigation of finite 3-groups G with commutator quotient G/G ′ ≃ R := C 3 × C 3 , represented by vertices of the descendant tree T (R), with the single restriction that the parent π(G) of G is metabelian. To this end, we first need parametrized presentations for all metabelian vertices of T (R). 9.1. 3-groups of coclass r = 1. The identification of 3-groups G with coclass cc(G) = 1, which are metabelian without exceptions [27], will be achieved with the aid of parametrized polycyclic power-commutator presentations, as given by Blackburn [9]: , 1} and w, z ∈ {−1, 0, 1} are bounded parameters, and the index of nilpotency m = cl(G) + 1 = cl(G) + cc(G) = log 3 (ord(G)) = lo(G) = n is an unbounded parameter. 9.2. 3-groups of coclass r ≥ 2. Metabelian 3-groups with coclass cc(G) ≥ 2 will be identified with the aid of parametrized polycyclic power-commutator presentations, given by Nebelung [7]: The flow of our investigations is guided by the present section concerning the remarkable infinite main trunk (P 2r−1 ) r≥2 of certain metabelian vertices in T which gives rise to the top vertices of all coclass forests (F (r)) r≥2 by periodic bifurcations and constitutes the germ of the newly discovered co-periodicity F (r + 2) ≃ F (r) of length two. Since the minimal possible values of the nilpotency class and logarithmic order of a finite metabelian 3-group with coclass cc(G) = r ≥ 2, belonging to the forest F (r), are given by c = cl(G) = r + 1 and lo(G) = cl(G) + cc(G) = 2r + 1, it follows that G must be an immediate descendant of step size s = 2 of its parent π(G) = G/γ c G. The crucial fact is that this parent is precisely the vertex π(G) = P 2r−1 with lo(π(G)) = 2r − 1 of the main trunk. In the following, we rather use the coclass j of the parent than r of the children. (1) In the descendant tree T (R) of the abelian root R := C 3 × C 3 = 9, 2 , there exists a unique infinite path of (reverse) directed edges (P 2j+1 ← P 2j+3 ) j≥1 such that, for each fixed coclass r = j + 1 ≥ 2, every metabelian 3-group G with G/G ′ ≃ (3, 3) and cc(G) = r is a proper descendant of P 2j+1 . (2) The trailing vertex P 3 is exactly the extra special Blackburn group G 3 0 (0, 0) = 27, 3 with exceptional transfer kernel type (TKT) a.1, κ = (0000).
11. Sporadic and periodic 3-groups G of even coclass cc(G) ≥ 4 Although formulated for the particular coclass r = 4, all results for periodic groups and most of the results for sporadic groups in this section are valid for any even coclass r ≥ 4. The only exception is the bigger (and thus pre-periodic) sporadic part F 0 (4) of the coclass forest F (4), described in Proposition 11.2, whereas the (co-periodic) standard case, the sporadic part F 0 (6) of the coclass forest F (6), is presented in Proposition 11.1. Figure 5 sketches an outline of the metabelian skeleton of the coclass forest F (4) in its top region. The vertices P 5 = 243, 3 ∈ F (2) and P 7 = 2187, 64 ∈ F (3), with the crucial bifucation from F (3) to F (4), belong to the infinite main trunk ( § 10). • 179 isolated vertices with dl = 3 and type b.10, • 23 capable vertices with dl = 3 and type b.10, whose children do not belong to F 0 (6), by definition.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung Although every branch contains 12 metabelian vertices, the primitive period length is ℓ = 2 rather than ℓ = 1, even for the metabelian skeleton, since the constitution 12 = 6 + 6 of branch B(10) is different from 12 = 4 + 8 for branch B(11), as proved above.
The claim of the virtual periodicity of branches has been proved generally for any coclass tree by du Sautoy [1], and independently by Eick and Leedham-Green [2]. Here, the strict periodicity was confirmed by computation up to branch B(31) and undoubtedly sets in at p * = 10.
The algebraic invariants of the groups represented by vertices forming the pre-period (B(9)) and the primitive period (B(10), B (11)) of the tree are given in Table 2. The leading six branches B(9), . . . , B (14) are drawn in Figure 6.
Remark 11.1. The algebraic information in Table 2 is visualized in Figure 6. By periodic continuation, the figure shows more branches than the table but less details concerning the exact order #Aut of the automorphism group.
Since C 1 (m n ) = 1 for all mainline vertices m n with n ≥ n * , according to Proposition 11.3, the unique capable child of m n is m n+1 , and each branch has depth dp(B(n)) = 1, for n ≥ n * . Consequently, the tree is also of depth dp(T ) = 1.
With the aid of Formula (5.9) in Theorem 5.1, the claims (2) and (4)  The algebraic invariants in Table 2, that is, depth dp, derived length dl, abelian type invariants of the centre ζ, relation rank µ, nuclear rank ν, abelian quotient invariants τ (1) of the first maximal subgroup, respectively τ 2 of the commutator subgroup, transfer kernel type κ, action flag σ, and the factorized order #Aut of the automorphism group have been computed by means of program scripts written for MAGMA [17].
Each group is characterized by the parameters of the normalized representative G m,n ρ (α, β, γ, δ) of its isomorphism class, according to Formula (9.2), and by its identifier 3 n , i in the SmallGroups Database [19].
The graph theoretic structure of the tree is determined uniquely by the numbers N 1 of immediate descendants and C 1 of capable immediate descendants of the mainline vertices m n with logarithmic order n = lo(m n ) ≥ n * = 9 and of capable vertices v with depth 1 and lo(v) ≥ n * + 1 = 10: (N 1 , C 1 ) = (13, 2) for mainline vertices m n with odd logarithmic order n ≥ 9, (N 1 , C 1 ) = (25, 3) for mainline vertices m n with even logarithmic order n ≥ 10, (N 1 , C 1 ) = (25, 0) for the capable vertex v of depth 1 and even logarithmic order lo(v) ≥ 10, (N 1 , C 1 ) = (13, 0) for two capable vertices v of depth 1 and odd logarithmic order lo(v) ≥ 11.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung [7, Thm. 5.1.16, pp. 178-179, and the fourth Figure,  The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [1] and [2]. Here, the strict periodicity was confirmed by computation up to branch B(30) and undoubtedly sets in at p * = 9. Proof. (of Theorem 11.3) Since every mainline vertex m n of the tree T has several capable children, C 1 (m n ) ≥ 2, but every capable vertex v of depth 1 has only terminal children, C 1 (v) = 0, according to Proposition 11.4, the depth of the tree is dp(T ) = 2. In this case, the cardinality of a branch B is the sum of the number N 1 (m n ) of immediate descendants of the branch root m n and the numbers N 1 (v i ) of terminal children of capable vertices v i of depth 1 with 2 ≤ i ≤ C 1 (m n ) (excluding the next mainline vertex v 1 = m n+1 ), according to Formula (5.10), that is, Applied to the primitive period, this yields #B(9) = 13 + 25 = 38, #B(10) = 25 + 13 + 13 = 51. According to Formula (5.14), the width of the tree is the maximum of all sums of the shape taken over all branch roots m n with logarithmic orders n * + 2 ≤ n ≤ n * + ℓ * + ℓ + 1. Applied to n * = 9, ℓ * = 0, and ℓ = 2, this yields wd(T ) = max(25 + 25, 13 + 13 + 13) = max(50, 39) = 50. Finally, we have IC(T ) = #B(9) + #B(10) = 38 + 51 = 89.  Table 3. In particular: (1) There are no groups with V 4 -action.  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 [17]. The other claims follow immediately from Table 3, continued indefinitely with the aid of the periodicity in Prop. 11.4.
The graph theoretic structure of the tree is determined uniquely by the numbers N 1 of immediate descendants and C 1 of capable immediate descendants of the mainline vertices m n with logarithmic order n = lo(m n ) ≥ n * = 9 and of capable vertices v with depth 1 and lo(v) ≥ n * + 1 = 10: (N 1 , C 1 ) = (13, 2) for mainline vertices m n with odd logarithmic order n ≥ 9, (N 1 , C 1 ) = (25, 3) for mainline vertices m n with even logarithmic order n ≥ 10, (N 1 , C 1 ) = (25, 0) for the capable vertex v of depth 1 and even logarithmic order lo(v) ≥ 10, (N 1 , C 1 ) = (13, 0) for two capable vertices v of depth 1 and odd logarithmic order lo(v) ≥ 11.
Theorem 11.7. (Strict isomorphism of the two trees.) Viewed as an algebraically structured infinite digraph, the second coclass-4 tree T 4 ( 2187, 64 − #2; 59) with mainline of type d.25 * in Figure 11 is strictly isomorphic to the first coclass-4 tree T 4 ( 2187, 64 − #2; 57) with mainline of type d.25 * in Figure 10. Only the presentations of corresponding vertices are different, but they share common algebraic invariants.   Proof. (Proof of Thm. 11.6 and Thm 11.7.) The claims have been verified with the aid of MAGMA [17] for all vertices V with logarithmic orders 9 ≤ lo(V ) ≤ 17. Pure periodicity of branches sets in with B(9) ≃ B (11). Thus, the claims for all vertices V with logarithmic orders lo(V ) ≥ 18 are a consequence of the periodicity theorems by du Sautoy in [1] and by Eick and Leedham-Green in [2], without the need of pruning the depth, which is bounded uniformly by 2.
12. Sporadic and periodic 3-groups G of odd coclass cc(G) ≥ 5 Although formulated for the particular coclass r = 5, all results on sporadic and periodic groups in this section are valid for any odd coclass r ≥ 5. The exemplary (co-periodic) sporadic part F 0 (5) of the coclass forest F (5) is presented in the following Proposition 12.1.
Theorem 12.1. The coclass-r forest F (r) with any odd r ≥ 5 is the disjoint union of its finite sporadic part F 0 (r) with total information content (12.1) s = #F 0 (r) = 207 and t = 4 infinite coclass-r trees T r (R r i ) with roots R r i := P 2r−1 − #2; n i , where (n i ) 1≤i≤4 = (25, 29, 37, 39) for r = 5. The algebraic invariants for groups with centre ζ = 1 2 , and in cumulative form for ζ = 1, are given for r = 5 in Table 6, where the parent vertex P 2r−1 = P 9 on the maintrunk is also included, but the 136 non-metabelian top vertices of depth dp = 0 are excluded.
The graph theoretic structure of the tree is determined uniquely by the numbers N 1 of immediate descendants and C 1 of capable immediate descendants of the mainline vertices m n with logarithmic order n = lo(m n ) ≥ n * = 11: (N 1 , C 1 ) = (30, 1) for the root m 11 with n = 11, (N 1 , C 1 ) = (24, 1) for all mainline vertices m n with even logarithmic order n ≥ 12, (N 1 , C 1 ) = (40, 1) for all mainline vertices m n with odd logarithmic order n ≥ 13.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung [7,Thm. 5.1.16,, and the third Figure,  The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [1] and [2]. Here, the strict periodicity was confirmed by computation up to branch B(33) and clearly sets in at p * = 12.
The algebraic invariants of the groups represented by vertices forming the pre-period (B(11)) and the primitive period (B(12), B(13)) of the tree are given in Table 7. The six leading branches B(11), . . . , B(16) are drawn in Figure 13.
Remark 12.1. The algebraic information in Table 7 is visualized in Figure 13. By periodic continuation, the figure shows more branches than the table but less details concerning the exact order #Aut of the automorphism group.
Since C 1 (m n ) = 1 for all mainline vertices m n with n ≥ n * , according to Proposition 12.2, the unique capable child of m n is m n+1 , and each branch has depth dp(B(n)) = 1, for n ≥ n * . Consequently, the tree is also of depth dp(T ) = 1.
Each group is characterized by the parameters of the normalized representative G m,n ρ (α, β, γ, δ) of its isomorphism class, according to Formula (9.2), and by its identifier 3 n , i in the SmallGroups Database [19].
The column with header # contains the number of groups with identical invariants (except the presentation), for each row. 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 [17]. The other claims follow immediately from Table 7, continued indefinitely with the aid of the periodicity in Proposition 12.2.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung [7,Thm. 5.1.16,, and the third Figure,  The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [1] and [2]. Here, the strict periodicity was confirmed by computation up to branch B(30) and certainly sets in at p * = 11.
(3) Depth, width, and information content of the tree are given by (12.3) dp(T 5 R 5 2 ) = 2, wd(T 5 R 5 2 ) = 75, and IC(T 5 R 5 2 ) = 75. The algebraic invariants of the vertices forming the root and the primitive period (B(11)) of the tree are presented in Table 8. The leading six branches B(11), . . . , B(16) are drawn in Figure 14. Table 8. Data for 3-groups G with 11 ≤ n = lo(G) ≤ 13 of the coclass tree T 5 R 5 2 # m, n ρ; α, β, γ, δ dp dl ζ µ ν τ (1) Proof. (Proof of Theorem 12.3) Since every mainline vertex m n of the tree T has three capable children, C 1 (m n ) = 3, but every capable vertex v of depth 1 has only terminal children, C 1 (v) = 0, according to Proposition 12.3, the depth of the tree is dp(T ) = 2. In this case, the cardinality of a branch B is the sum of the number N 1 (m n ) of immediate descendants of the branch root m n and the numbers N 1 (v i ) of terminal children of capable vertices v i of depth 1 with 2 ≤ i ≤ C 1 (m n ) (excluding the next mainline vertex v 1 = m n+1 ), according to Formula (5.10), that is, Applied to the primitive period, this yields #B(11) = 25 + 25 + 25 = 75. According to Formula (5.14), the width of the tree is the maximum of all sums of the shape taken over all branch roots m n with logarithmic orders n * + 2 ≤ n ≤ n * + ℓ * + ℓ + 1. Applied to n * = 11, ℓ * = 0, and ℓ = 1, this yields wd(T ) = max(25 + 25 + 25) = max(75) = 75. Finally, we have IC(T ) = #B(11) = 75.

Corollary 12.2. (Actions and relation ranks.)
The algebraic invariants of the vertices of the structured coclass-5 tree T 5 R 5 2 are listed in Table 8. In particular: (1) There are no groups with GI-action, let alone with RI-or V 4 -action.
(2) The relation rank is given by µ = 5 for the mainline vertices 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 [17]. The other claims follow immediately from Table 8, continued indefinitely with the aid of the periodicity in Proposition 12.3. Figure 15. The unique coclass-5 tree T 5 (P 9 − #2; 37) with mainline of type d.    The branches B(i), i ≥ n * = 11, of the unique coclass-5 tree T 5 (P 9 − #2; 37) with mainline vertices of transfer kernel type d.23 * , κ ∼ (0243), are purely periodic with primitive length ℓ = 1 and without pre-period, ℓ * = 0, that is, B(i + 1) ≃ B(i) are isomorphic as structured digraphs, for all i ≥ p * = n * + ℓ * = 11. The graph theoretic structure of the tree is determined uniquely by the numbers N 1 of immediate descendants and C 1 of capable immediate descendants for mainline vertices m n with logarithmic order n = lo(m n ) ≥ n * = 11 and for capable vertices v with depth 1 and lo(v) ≥ n * + 1 = 12: (N 1 , C 1 ) = (15, 2) for all mainline vertices m n of any logarithmic order n ≥ 11, (N 1 , C 1 ) = (25, 0) for the capable vertex v of depth 1 and any logarithmic order lo(v) ≥ 12.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung [7, Thm. 5.1.16, pp. 178-179, and the third Figure,  The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [1] and [2]. Here, the strict periodicity was confirmed by computation up to branch B(30) and certainly sets in at p * = 11. The algebraic invariants of the vertices forming the root and the primitive period (B(11)) of the tree are presented in Table 9. The leading six branches B(11), . . . , B(16) are drawn in Figure 15.
The results concerning the metabelian skeleton confirm the corresponding statements in the dissertation of Nebelung [7, Thm. 5.1.16, pp. 178-179, and the third Figure,  The claim of the virtual periodicity of branches has been proved generally for any coclass tree in [1] and [2]. Here, the strict periodicity was confirmed by computation up to branch B(30) and certainly sets in at p * = 11. (3) Depth, width, and information content of the tree are given by (12.5) dp(T 5 R 5 4 ) = 2, wd(T 5 R 5 4 ) = 45, and IC(T 5 R 5 4 ) = 85. The algebraic invariants of the vertices forming the root and the primitive period (B(11), B(12)) of the tree are presented in Table 10. The leading six branches B(11), . . . , B(16) are drawn in Figure  16.
Proof. (Proof of Theorem 12.5) Since every mainline vertex m n of the tree T has several capable children, C 1 (m n ) ≥ 2, but every capable vertex v of depth 1 has only terminal children, C 1 (v) = 0, according to Proposition 12.5, the depth of the tree is dp(T ) = 2. In this case, the cardinality of a branch B is the sum of the number N 1 (m n ) of immediate descendants of the branch root m n and the numbers N 1 (v i ) of terminal children of capable vertices v i of depth 1, 2 ≤ i ≤ C 1 (m n ) (where v 1 is the next mainline vertex and must be discouraged), according to Formula (5.10), Applied to the primitive period, this yields #B(11) = 15 + (15 + 15) = 45 and #B(12) = 15 + 25 = 40. According to Formula (5.14), the width of the tree is the maximum of all sums of the shape #Lyr n T = #{v ∈ T | lo(v) = n} = N 1 (m n−1 ) + taken over all branch roots m n with logarithmic orders n * + 2 ≤ n ≤ n * + ℓ * + ℓ + 1. Applied to n * = 11, ℓ * = 0, and ℓ = 2, this yields wd(T ) = max (  The algebraic invariants of the vertices of the structured coclass-5 tree T 5 R 5 4 are listed in Table  10. In particular: (1) There are no groups with GI-action, let alone with RI-or V 4 -action.
Since C 1 (m n ) = 1 for all mainline vertices m n with n ≥ n * , according to Proposition 13.1, the unique capable child of m n is m n+1 , and each branch has depth dp(B(n)) = 1, for n ≥ n * . Consequently, the tree is also of depth dp(T ) = 1.
The algebraic invariants in Table 11, that is, defect of commutativity k, depth dp, derived length dl, abelian type invariants of the centre ζ, relation rank µ, nuclear rank ν, abelian quotient invariants τ (1) of the first maximal subgroup, respectively τ 2 of the commutator subgroup, transfer kernel type κ, action flag σ, and the factorized order #Aut of the automorphism group have been computed by means of program scripts written for MAGMA [17].
Each group is characterized by the parameters of the normalized representative G n a (z, w) of its isomorphism class, according to Formula (9.1), and by its identifier 3 n , i in the SmallGroups Database [19].
The column with header # contains the number of groups with identical invariants (except the presentation and identifier), for each row. The algebraic invariants of the vertices of the structured coclass-1 tree T 1 R with abelian root R = 9, 2 ≃ C 3 × C 3 , which is drawn in Figure 17, are listed in Table 11. In particular: (1) The groups with V 4 -action are the root R, all mainline vertices G n 0 (0, 0), n ≥ 3, and the terminal vertices G n 0 (±1, 0) with even logarithmic order n ≥ 4. (2) With respect to the transfer kernel types, all mainline groups of type a.1 * , κ = (0000), and the leaves of type a.3, κ ∼ (2000), with odd class c = n − 1 ≥ 3, possess a V 4 -action. (3) The relation rank is given by µ = 4 for the mainline vertices G n 0 (0, 0), n ≥ 3, µ = 2 for the terminal extraspecial group G 3 0 (0, 1), and µ = 3 otherwise. (4) All terminal vertices with odd class, and the mainline vertices with even class, possess an RI-action. The terminal vertices with odd class are Schur+1 σ-groups [29,30].
Proof. (of Corollary 13.1) 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 [17]. The other claims follow immediately from Table 11, continued indefinitely with the aid of the periodicity in Proposition 13.1. A Schur+1 σ-group has an RI-action and relation rank µ ≤ 3 [29,30].

Conclusion
In the core sections 11 and 12 of this paper, we have elaborated our long desired proof that the pruned tree of all finite 3-groups with elementary bicyclic commutator quotient, which do not arise as descendants of non-metabelian groups, can be described with a finite amount of data. for r = 1, 739 = 515 + 224 for r = 4, 501 = 207 + 294 for each odd r ≥ 5, 581 = 357 + 224 for each even r ≥ 6.
Proof. The total information content of a coclass forest is the sum of the cardinality of its sporadic part F 0 (r) and the information contents of its pairwise non-isomorphic coclass trees T r R r i IC(F (r)) = #F 0 (r) + i IC(T r R r i ).
Due to the exceptional complexity of the pre-periodic forests F (2) and F (3), their information contents are unknown up to now. However, since they are certainly finite, this does not obfuscate our clear and beautiful results concerning the infinitely many co-periodic forests F (r) with r ≥ 4, which can be reduced to the finite information content of the primitive co-period (F (5), F (6)).