On the classification of groups of prime-power order by coclass: The 3-groups of coclass 2

On the classification of groups of prime-power order by coclass: The 3-groups of coclass 2

Bettina Eick, C.R. Leedham-Green, M.F. Newman and E.A. O’Brien

Abstract

In this paper we take a significant step forward in the classification of 3-groups of coclass 2. Several new phenomena arise. Theoretical and computational tools have been developed to deal with them. We identify and are able to classify an important subset of the 3-groups of coclass 2. With this classification and further extensive computations, it is possible to predict the full classification.

On the basis of the work here and earlier work on the p-groups of coclass 1, we formulate another general coclass conjecture. It implies that, given a prime p and a positive integer r, a finite computation suffices to determine the p-groups of coclass r.

1 Introduction

The coclass of a group of order pⁿ and nilpotency class cis defined as n−c. In 1980, Leedham-Green & Newman [16] made a series of conjectures about finite p-groups, using coclass as the primary invariant. A detailed account of the proofs of these conjectures, and the resultant program of study, can be found in [14].

The goal is to classifyp-groups via coclass. We expect that it is possible to reduce the classification to a finite calculation, and that the p-groups of a given coclass can be partitioned into finitely many families, where the groups in a family share similar structure and can be described by a parametrised presentation. One approach to achieving this goal is to understand the structure of the coclass graph G(p, r). Its vertices are the p-groups of coclass r, one for each isomorphism type, and its edges are P →Q, with Qisomorphic to the quotient P/L_c(P), where L_c(P) is the last non- trivial term of the lower central series of P. If G(p, r) can be constructed from a finite subgraph using a finite number of graph-theoretic operations, then this may assist in realising our goal.

We thank Heiko Dietrich for his careful reading, many comments, and for an illustration; and Marcus du Sautoy for useful discussions. Both Eick and O’Brien were partially supported by the Alexander von Humboldt Foundation. All authors were partially supported by the Marsden Fund of New Zealandvia grant UOA1015.

Eick & Leedham-Green [11] proved that the graph for the 2-groups of a given coclass can be constructed from a finite subgraph by applying just one type of operation to the subgraph – and this operation has an analogue at the group-theoretic level. That the graph exhibits such a simple structure was first conjectured by Newman & O’Brien [18]. Their Conjecture P was proved by du Sautoy [7] and in a much sharper form in [11]. The results of [11] have already been applied to study the automorphism groups of 2-groups [9], and Schur multiplicators of p-groups [10].

Blackburn’s classification [3] of the 3-groups of coclass 1 implies that these groups exhibit behaviour similar to that proved for 2-groups. But we know from other in- vestigations that the results of [11] are not generally true. The 5-groups of coclass 1 have been investigated in [4, 5, 6, 15, 17]; while this work suggests that G(5,1) can be constructed from a finite subgraph, the above operation does not suffice.

The number of isomorphism classes of p-groups of coclass r of order pⁿ, for odd p, is bounded by a linear function of n precisely when (p, r) is one of (3,1), (3,2) or (5,1). We consider here the 3-groups of coclass 2. The study ofG(3,2) goes back to the late 1970s (see [1, 2]) and early results played a role in the development of the original coclass conjectures. Our computations, reported in Section 8, show that the complete graph is very dense. In Theorems 5.10 and 7.1 we determine a significant subgraph: the skeleton graph (defined in Section 3.4). While this subgraph is comparatively sparse, it exhibits the broad structure ofG(3,2). Our computations suggest that the complete structure of G(3,2) can be determined from a finite subgraph.

The skeleton graph of G(3,2) exhibits some new features; we consider these in Section 7. Its determination required dealing with number-theoretic problems similar to those considered by Leedham-Green & McKay [15] in their investigation of skeleton graphs of G(p,1) for p ≥ 5. That G(3,2) does not reveal all complexities that arise in classifying p-groups by coclass is demonstrated by Dietrich [6]. We conclude by stating Conjecture W: a new conjecture about the graph-theoretic operations needed to describe G(p, r) for arbitrary p and r.

We briefly consider its implications for the goal of classifying the p-groups of co- class r, one already realised via Theorem P for the prime 2. A constructive proof of ConjectureW, its analogue for odd primes, that provides explicit bounds would reduce this classification for a fixed p and r to a finite calculation. It would also allow us to determine a recursive formula in n for the number of isomorphism types of groups of order pⁿ and coclass r.

2 Preliminaries

2.1 Coclass trees

By [14, Corollary 7.4.13], every infinite pro-p-groupGof coclassr is ap-adic pre-space group. Namely, Ghas a normal subgroup T which is a free, finitely generated module over the ringZp ofp-adic integers, andQ:=G/T is a finitep-group that acts uniserially onT. While T is not unique, the rank d ofT asZp-module is an invariant of Gcalled

its dimension; it is a consequence of [14, Corollary 7.4.13 and Theorem 10.5.12] that d= (p−1)p^s for some s ∈ {0, . . . , r−1}.

The uniserial action implies that the subgroups defined by T₀ := T and T_i+1 :=

[T_i, Q] form a chain T = T₀ > T₁ > . . . > T_i > T_i+1 > . . . > {0} with [T_i : T_i+1] = p and T_i+d = pT_i for 0 ≤ i < ∞. We set T_∞ := {0}. This chain extends to a doubly infinite series · · · > T₋₂ > T₋₁ > T₀ > T₁ > T₂ >· · · and again T_i+1 has index p in T_i for all i.

IfP and Qare groups inG(p, r), thenP is adescendantof Qif there is a (possibly trivial) path in G(p, r) from P to Q. The descendant tree of Q is the subtree of its descendants, and has root Q.

If G is an infinite pro-p-group of coclass r, then G/L_i+1(G), the quotient of G having class i, is a finite p-group of coclass at most r for all i > 0, and the coclass of G/L_i+1(G) is preciselyr for all but finitely many values ofi. Moreover, since there are only finitely many infinite pro-p-groups of coclassrup to isomorphism [14, p. (viii)], for sufficiently large i the group G/L_i+1(G) is a quotient of only one infinite pro-p-group of coclass r. Chooseiminimal with respect to these properties. The coclass treeT(G) is the descendant tree of G/L_i+1(G) in G(p, r).

There are only finitely many coclass trees in G(p, r) and only finitely many groups in G(p, r) are not contained in a coclass tree [13, Proposition 2.2]. Hence the study of the broad structure of G(p, r) reduces to an investigation of its coclass trees.

2.2 Mainline and branches

LetGbe an infinite pro-p-group of coclass r, and letT(G) be its coclass tree with root G/L_i+1(G). The quotients G/L_i+1(G), G/L_i+2(G), . . . form a unique maximal infinite path, or mainline, in T(G).

For j ≥ i, let Bj denote the subtree of T(G) consisting of G/L_j+1(G) and all of its descendants that are not descendants of G/L_j+2(G). Thus B^j is a finite subtree of T(G), and is its jth branch. Hence T(G) consists of an infinite sequence of trees Bi,Bi+1, . . . , connected by the mainline. The subtree of all vertices in Bj of distance at most k from G/L_j+1(G) is denoted by B^j,k.

2.3 Periodicity

Eick & Leedham-Green [11, Theorem 29] prove the following.

Theorem 2.1 Let G be an infinite pro-p-group of coclass r and dimension d. There exists an explicit function f such that, for every positive integerk and every j ≥f(k), there is a graph isomorphism π_j :Bj,k → Bj+d,k.

We say that T(G) has period d and defect f; bounds for the latter appear in [11].

This theorem suggests that we arrange the infinitely many branches of a coclass tree T(G) with root G/L_i+1(G) into d sequences (B^i+e,B^i+e+d,B^i+e+2d, . . .) for 0≤e < d.

The depth of a rooted tree is the length of a maximal path from a vertex to the root. A sequence of branches has bounded depth if the depths of its trees B^i+e+kd are bounded by a constant. (If every sequence of branches has bounded depth, thenT(G) has bounded depth.) Theorem 2.1 implies that a sequence of branches of bounded depth is ultimately constant, and can therefore be constructed from a finite subsequence.

Every sequence of branches of a coclass tree inG(2, r) or G(3,1) has bounded depth (see [14, Theorem 11.4.4]). In these cases Theorem 2.1 shows that Bj+d ∼= Bj for large enough j. The proof in [11] of Theorem 2.1 is underpinned by an explicit group- theoretic construction. It defines families of p-groups of coclass r where the groups in a family share similar structure and are described by a parametrised presentation.

All coclass graphs other thanG(2, r) andG(3,1) contain coclass trees of unbounded depth (see [14, Theorem 11.4.4]) and so are not covered by Theorem 2.1. We show that both types of coclass trees occur in G(3,2).

2.4 Notation

Much of our notation is standard. For consistency, if G is the split extension AnB or the non-split extension A·B, then in both cases B is normal in G. We denote a term of the lower central series ofGbyL_i(G) fori >0; and a left-normed commutator [a, b, . . . , b

| {z }

i

] by [a,_ib].

3 Skeletons

In this section we recall a construction by Leedham-Green & McKay [14, §8.4] that is central to the investigation of branches of unbounded depth. Throughout this section, let pbe an odd prime.

LetG be an infinite pro-p-group of coclass r. Recall that Gis an extension of a d- dimensionalZp-moduleT by a finite p-group Qwhich acts uniserially onT with series T = T₀ > T₁ > . . .. The exterior square T ∧T is a ZpQ-module under the diagonal action of Q. If i < j then we defineT_i∧T_j =T_j∧T_i to be theZp-submodule ofT_i∧T_i spanned by {s∧t |s ∈T_i, t∈T_j}.

3.1 Twisting homomorphisms

Let γ :T ∧T →T be a ZpQ-module homomorphism. Then γ(T<sub>`</sub>∧T) is a Q-invariant subgroup of T for every `≥0. Let γ(T ∧T) =T_j for j ≥0, and letγ(T_j∧T) =T_k. If j ≤m ≤k, then γ induces a homomorphism γ_m :T /T_j∧T /T_j →T_j/T_m defined by

γ_m(a+T_j ∧b+T_j) =γ(a∧b) +T_m.

This induced homomorphism can be used to define a new group multiplication ‘·’ on T /T_m that turns the additive abelian group T /T_m into a multiplicative group of class

at most 2. More precisely, for a, b∈T we define

(a+T_m)·(b+T_m) = (a+b+T_m) + ¹₂γ_m(a+T_j ∧b+T_j).

The resulting groupT_γ,m := (T /T_m,·) has order p^m. Commutators are evaluated easily in T_γ,m as

[a+T_m, b+T_m] =γ_m(a+T_j ∧b+T_j).

Ifm =j, then γ_m is the trivial homomorphism, and T_γ,m is abelian. If j < m≤ k then T_γ,m has derived subgroupT_j/T_m and class precisely 2. Also T_γ,n is a quotient of T_γ,m if j ≤n ≤m.

Lemma 3.1 With the above notation, let γ(T ∧T) =T_j and γ(T_j ∧T) =T_k, and let d be the rank of T, as ZpQ-module.

(a) If j is infinite, or equivalently γ = 0, then m is infinite and T_γ,∞ ∼= (T,+).

(b) If j is finite, or equivalently γ 6= 0, then 2j−d < k≤2j+d.

(c) If j ≤m≤k, then T_pⁱ_γ,m+2id is defined for every i≥0.

Proof:

(a) This follows directly from the definition.

(b) Writej =id+ewith 0≤e < d. ThenT_k=γ(T_j∧T) =γ(pⁱT_e∧T) =pⁱγ(T_e∧T), and T_j+d = pT_j = γ(pT ∧T) ≤ γ(T_e ∧T) ≤ γ(T ∧T) = T_j. Hence pⁱT_j+d ≤ T_k ≤ pⁱT_j or, equivalently, id+j +d ≥ k ≥ id+j. As id = j −e, this yields 2j+ (d−e)≥k≥2j−e.

(c) Note that pⁱγ(T ∧T) =pⁱT_j =T_j+id, and

pⁱγ(T_j+id∧T) =p²ⁱγ(T_j ∧T) =p²ⁱT_k=T_k+2id.

Thus if j ≤m≤k, then j+ 2id≤m+ 2id≤k+ 2id, and the result follows. •

3.2 Skeleton groups

Assume that G splits over T. Let γ : T ∧T → T be a ZpQ-module homomorphism, whereγ(T∧T) =T_j andγ(T_j∧T) =T_k, andj ≤m≤k < ∞. Since the natural action of Q onT /T_m respects the new multiplication induced by γ, we can define a skeleton groupG_γ,m :=QnT_γ,m. Ifj is sufficiently large, thenG_γ,m is a group of depthm−j in the branch of T(G) with rootQnT /T_j. Lemma 3.1(c) shows that the homomorphism pⁱγ for i≥0 defines a skeleton groupG_pⁱ_γ,m+2id of depthm−j+idin the branch with rootQnT /T_j+id. Thus the sequence of branches with roots QnT /T_j+idfori= 0,1, . . . has unbounded depth.

Now assume thatG is a non-split extension ofT byQ. As described in [14,§10.4], there exists a unique minimal supergroup S of T such that G embeds in the infinite pro-p-group H :=QnS of finite coclass. A finite upper bound to [H : G] = [S : T] is given in [14, Theorem 10.4.6]. Let γ :S∧S →S be aZpQ-module homomorphism where γ(S ∧S) = S_j and γ(S_j ∧S) = S_k, and j ≤ m ≤ k. Now H_γ,m = QnS_γ,m is the skeleton group defined by γ and m. Assume that the largest mainline quotient of H_γ,m has class j, so H/L_j+1(H) ∼= H_γ,m/L_j+1(H_γ,m). Assume also that j is large enough so that L_j+1(H)≤G. DefineG_γ,m as the full preimage in H_γ,m ofG/L_j+1(H).

Then G_γ,m is the skeleton group forG defined by γ and m.

Lemma 3.2 Every constructible group in the sense of [14, Definition 8.4.9] is a skele- ton group, and conversely.

Proof: This is straightforward if the infinite pro-p-groupGsplits overT. IfGis a non- split extension ofT byQ, then a constructible groupG_αforGis defined as an extension determined by α ∈ Hom_Q(T ∧T, T) in [14, Definition 8.4.9]. This homomorphism extends to the minimal split supergroup S of T and defines a constructible group G_α for the pro-p-groupQnS. SinceG_α is a skeleton group for QnS and containsG_α as an appropriately embedded subgroup, G_α is a skeleton group. •

3.3 The isomorphism problem for skeleton groups

Assume thatT is a characteristic subgroup ofG. (This assumption is always satisfied in our later applications.) Since T_i fori≥0 is then characteristic inG, each α∈Aut(G) induces an automorphism of T /T_i. Hence we can define an action of Aut(G) on the set of homomorphisms γ_m induced by surjections γ ∈ Hom_Q(T ∧T, T_j). Namely, for x, y ∈T /T_j let

α(γ_m)(x∧y) := α(γ_m(α⁻¹(x)∧α⁻¹(y))).

Lemma 3.3 Let γ and γ⁰ be two surjections in Hom_Q(T ∧T, T_j), and assume that there exists α∈Aut(G) with α(γ_m) =γ_m⁰ . Then G_γ,m∼=G_γ⁰_,m.

Proof: First consider the case where G=QnT. Since T is characteristic in G, the automorphism α induces automorphisms of Q and of T. The restriction of α to T is a Zp-linear map. Hence for a, b∈T, if ·γ and ·γ⁰ denote the twisted operations on T /T_m defined by γ and γ⁰ respectively then

α((a+T_m)·^γ(b+T_m)) = α((a+b+T_m) + ¹₂γ_m(a+T_j∧b+T_j))

= α(a+T_m) +α(b+T_m) + ¹₂α(γ_m(a+T_j∧b+T_j))

= α(a+T_m) +α(b+T_m) + ¹₂γ_m⁰ (α(a+T_j)∧α(b+T_j)))

= α(a+T_m)·γ⁰ α(b+T_m).

Thus α induces an isomorphism T_γ,m → T_γ⁰_,m. Since G = QnT, we deduce that G_γ,m =QnT_γ,m and the mapG_γ,m →G_γ⁰_,m: (g, x)7→(α(g), α(x)) is an isomorphism.

Now suppose that G is a non-split extension of T by Q, and let H = GS be a minimal split supergroup. An automorphism of G restricts to an automorphism of T and this, in turn, extends uniquely to S. Since H = GS, an automorphism of G extends to an automorphism of H which normalises G. The split case implies that

H_γ,m ∼=H_γ⁰_,m and thusG_γ,m ∼=G_γ⁰_,m. •

The isomorphisms induced by this action of Aut(G) on skeleton groups areorbit iso- morphisms. The determination of the orbit isomorphisms is an important step towards a solution of the isomorphism problem for the skeleton groups. Other isomorphisms can arise, as the study of 3-groups of coclass 2 shows. We call them exceptional. Their complete determination requires considerable care.

3.4 The skeleton graph

Let P be a skeleton group in T(G) of class c, and let L_c(P) be the last non-trivial term of its lower central series. If P/L_c(P) is in T(G), then it is also a skeleton group.

Thus the skeleton groups define a subgraph, theskeleton graph, ofT(G) which includes the mainline. The subtree of branch B^j consisting of skeleton groups defines S^j, the skeleton of Bj.

ThetwigofP is the subtree of all descendants ofP that are not descendants of any skeleton group that is a proper descendant of P. Thus T(G) is partitioned into twigs, and the twigs are connected by the skeleton graph of T(G).

The following is a consequence of [14, Theorem 11.3.9] and Lemma 3.2.

Theorem 3.4 There is an absolute bound to the depth of the twigs in T(G).

Hence the skeleton graph exhibits the broad structure of T(G), the twigs contain the fine detail. In particular, there are only finitely many isomorphism types of twigs.

Conjecture W (Section 9) suggests that there are patterns in the isomorphism types of twigs which occur in a coclass tree.

4 The infinite pro-3-groups of coclass 2

We show that there are 16 infinite pro-3-groups of coclass 2 up to isomorphism, and identify the four coclass trees in G(3,2) that have unbounded depth.

Theorem 4.1 There are six isomorphism types of infinite pro-3-groups of coclass 2 with non-trivial centre. They have the following pro-3 presentations:

{a, t, z|a³ =z^f,[t, t^a] =z^g, tt^at^a2 =z^h, z³ = [z, a] = [z, t] = 1} where (f, g, h) is one of (0,0,0),(0,0,1),(0,1,0),(0,1,2),(1,0,0), or (1,1,2).

Proof: Every infinite pro-3-group of coclass 2 with non-trivial centre is a central extension of the cyclic group of order 3 by the (unique) infinite pro-3-groupSof coclass 1 (see [14, §7.4]). This is reflected in the presentations, sincehzi is a central subgroup of order 3 with quotient S. The isomorphism types of infinite pro-3-groups of coclass 2 with non-trivial centre correspond one-to-one to the orbits of Aut(S)×Aut(Z/3Z)

on H²(S,Z/3Z)∼= (Z/3Z)³. •

Every infinite pro-3-groupG of coclass 2 with trivial centre is a 3-adic space group of dimensiond= 2·3^s wheres∈ {0,1}. As the unique 3-adic space group of dimension 2 has coclass 1, it follows that d = 6. Thus G is an extension of a free Z3-module of rank 6 by a finite 3-group acting faithfully and uniserially on the module.

We use number theory to describe the infinite pro-3-groups of coclass 2 in more detail. Let K be the ninth cyclotomic number field. Then K =Q3(θ), whereQ3 is the field of 3-adic numbers, and θ is a primitive ninth root of unity; so 1 +θ³ +θ⁶ = 0.

The ring of integers O of K is a free Z3-module of rank 6 generated by 1, θ, . . . , θ⁵. Let W be the group of Z3-linear maps of O generated by the permutations a = (1, θ, . . . , θ⁸) andy= (1, θ³, θ⁶). Then W has order 81 and is isomorphic to the wreath product of two cyclic groups of order 3. Further, b = (θ, θ⁴, θ⁷)(θ², θ⁸, θ⁵) = a⁵y²ay, and so b ∈ W. The action of W on O extends to a Q3-linear action on K. We write k^w for the image of k ∈ K under w ∈ W. Note that k^a = kθ, and thus a acts as multiplication by θ. Further, k^b = σ₄(k), where, for i prime to 3, the map σ_i is the Galois automorphism of K defined by θ 7→θⁱ.

The split extension W n(O,+) = {(w, o) | w ∈ W, o ∈ O} is a uniserial 3-adic space group of coclass 4 with translation subgroup {(1, o) | o ∈ O} ∼= (O,+). Let p= (θ−1)O denote the unique maximal ideal in O. The maximal W-invariant series in the translation subgroup (O,+) is O=p⁰ >p¹ >p² > . . ..

It is sometimes useful to have a multiplicative version of (O,+). We denote this multiplicative group by T and write T_i for the subgroup corresponding topⁱ. If t₀ ∈T corresponds to 1∈ O, thent_i = [t₀,_ia]∈T_i\T_i+1 corresponds to (θ−1)ⁱ ∈pⁱ\pⁱ⁺¹. In the multiplicative setting we write the split extension W nT as {wt|w∈W, t∈T}.

Every 3-adic space group of coclass 2 embeds as a subgroup of finite index inWnT. Define C:=hai and D:=ha, bi as subgroups of W, and so ofW nT.

Theorem 4.2 There are ten isomorphism types of infinite pro-3-groups of coclass 2 with trivial centre. They are:

(a) R :=ha, t₀i so R =CnT.

(b) G_i :=ha, bt_i−1i fori= 1,2,3, so G_i =D·T_i and has index 3 in its minimal split supergroup H_i−1 :=ha, b, t_i−1i.

(c) ha, yt_i−2i and ha, ya⁻¹t_i−2i for i= 2,3,4. Each subgroup is isomorphic to W ·T_i and has index 9 in W nT_i−2.

That there are precisely ten isomorphism types follows from [8]; the algorithm given there can be used to obtain descriptions of the groups.

The following can be deduced from Corollary 11.4.2 and Theorem 11.4.4 of [14].

Theorem 4.3 The six groups of Theorem4.1and the six groups in part(c)of Theorem 4.2 have coclass trees whose sequences of branches all have bounded depth.

Since the trees of bounded depth are covered by Theorem 2.1, we do not investigate them further. As we prove below, the remaining groups – those in parts (a) and (b) of Theorem 4.2 – have coclass trees with unbounded depth, and so we study them.

In Sections 5 and 7 respectively, we study T(R) and T(G_i) for 1 ≤ i ≤ 3; in Section 6 we prepare for the latter by studying T(H_i) for 0 ≤ i ≤ 2. We first theoretically determine their skeleton graphs and then investigate their twigs.

5 The skeleton groups in T (R)

The root of T(R) has order 3⁹. LetBj denote the branch ofT(R) whose root has class j and order 3^j+2, and let S^j be the skeleton of B^j.

In this section we determine Sj for every j ≥ 7. The general construction of skeleton groups is described in Section 3. In number-theoretic notation, it uses the surjections γ ∈Hom_C(O ∧ O,p^j), and their imagesγ_m in Hom_C(O/p^j∧ O/p^j,p^j/p^m), for j ≤ m ≤ k, where k is as defined in Section 3.1; we determine the value of k in Lemma 5.3. The resulting skeleton groups R_γ,m := CnT_γ,m have order 3^m+2, class m, and depth m−j in S^j. We first describe Hom_C(O ∧ O,O), then we determine all orbit isomorphisms, and finally show that there are no exceptional isomorphisms. We also give presentations for the skeleton groups and describe their automorphisms. To facilitate the determination of S^j, we use a different representation ofR for each j.

5.1 The homomorphism space

We now describe the space of homomorphisms which determine the skeleton groups in T(R). Recall that σ_i is the Galois automorphism of K defined by θ 7→ θⁱ where i is prime to 3. We first define the map

ϑ :O ∧ O → O:x∧y 7→σ₂(x)σ₋₁(y)−σ₋₁(x)σ₂(y). (1) Lemma 5.1 The map ϑ is an element of Hom_C(O ∧ O,O). If i and j are non- negative integers, then ϑ maps pⁱ ∧p^j onto p^i+j+ε, where ε = 3 if i ≡ j mod 3, and ε= 2 otherwise.

Proof: Clearly ϑ(θx∧θy) = σ₂(θx)σ₋₁(θy)−σ₋₁(θx)σ₂(θy) = θϑ(x∧y). Henceϑ is compatible with the action of θ, and thusϑ∈Hom_C(O ∧ O,O). The image of pⁱ∧p^j

underϑ is generated byϑ((θ−1)ⁱθ^u1∧(θ−1)^jθ^u2) for 0≤u₁, u₂ ≤5, aspⁱ = (θ−1)ⁱO and Ois generated by 1, θ, . . . , θ⁵ as aZ3-module. Lete=i−j andf =u₁−u₂. Then

ϑ((θ−1)ⁱθ^u1 ∧(θ−1)^jθ^u2)

= (θ²−1)^j+eθ^2(u2+f)(θ⁻¹−1)^jθ^−u2 −(θ⁻¹−1)^j+eθ^−(u2+f)(θ² −1)^jθ^2u2

= (θ²−1)^j(θ⁻¹−1)^j(θ−1)^eθ^u2−f−e[(1 +θ)^eθ^3f+e−(−1)^e]

= (θ−1)^i+jc(i, j, u₁, u₂) [(1 +θ)^eθ^3f+e−(−1)^e],

where c(i, j, u₁, u₂) is a unit. Consider the term (1 +θ)^eθ^3f+e−(−1)^e: if e≡0 mod 3 then it is in p³; if e ≡ 0 mod 3 and f 6≡ 0 mod 3 then it is in p³\p⁴; if e 6≡ 0 mod 3

then it is in p²\p³. •

Let U denote the unit group of the ring O.

Lemma 5.2 Let i and j be non-negative integers.

(a) Hom_C(pⁱ∧pⁱ,p^i+j) ={c(θ−1)^j−i−3ϑ|c∈ O}=p^j−i−3ϑ.

(b) Hom_C(pⁱ∧pⁱ,p^i+j)\Hom_C(pⁱ∧pⁱ,p^i+j+1) = (θ−1)^j−i−3Uϑ.

Proof: Lemma 5.1 shows that (θ −1)^j−i−3ϑ is a surjective element of Hom_C(pⁱ ∧ pⁱ,p^i+j). By [14, Theorem 11.4.1], the set{ϑ}is aK-basis of Hom_C(K∧K, K). Hence Hom_C(pⁱ∧pⁱ,p^i+j) =p^j−i−3ϑ for every j ≥0. Also p^l\p^l+1 = (θ−1)^lU for every l. •

5.2 A change of representation and notation

To describeSj, the skeleton ofBj, we must determine the orbits of the action of Aut(R) on the set of homomorphisms induced by the surjections of Lemma 5.2 for i= 0. The term (θ−1)^j−3 introduces technical complications to these computations. We avoid these by adjusting our notation.

Let R_h = C nT_h ∼= C np^h for h ≥ 0. Then R₀ = R and R_h ∼= R for h ≥ 1.

Using R_j−3 instead of R₀, the skeleton groups in B^j correspond to the Z3C-module surjections

γ :p^j−3∧p^j−3 →p^2j−3.

Lemma 5.2 shows that these surjections can be written as cϑ for some unit c ∈ U, avoiding (θ−1)^j−3.

Ifj ≤m ≤k, then the new surjection γ induces the homomorphism γ_m :p^j−3/p^2j−3∧p^j−3/p^2j−3 →p^2j−3/p^m+j−3.

As with the previous notation, γ_m defines a multiplication ‘·’ on the set p^j−3/p^m+j−3. We denote the resulting group (p^j−3/p^m+j−3,·) byT_j−3,γ,m. It has order 3^m and derived subgroup p^2j−3/p^m+j−3. NowR_j−3,γ,m=CnT_j−3,γ,mis a skeleton group inS^j of order 3^m+2, class m, and depthm−j.

Lemma 5.3 For every j ≥ 7, the skeleton Sj has depth j − χ_j, where χ_j = 0 if 3 divides j, and χ_j = 1 otherwise. In particular, every skeleton S^j is non-trivial, and every sequence of branches in T(R) has unbounded depth.

Proof: Lemma 5.1 implies that ϑ maps p^j−3∧p^j−3 onto p^2j−3, andp^2j−3∧p^j−3 onto p^3j−6+ε, where ε= 3 if 3 divides j, and ε = 2 otherwise. HenceR_j−3,γ,m is defined for

j ≤m ≤2j −3 +ε= 2j−χ_j. •

Hence, for the remainder of Section 5, we assume thatj ≥7 andj ≤m≤2j−χ_j, where χ_j is defined in the above lemma.

5.3 The automorphism group of R

We construct the automorphism group of R_h for h≥0 from three subgroups.

Galois automorphisms. Observe that σ₂ generates the Galois group Gal(K,Q3), which is cyclic of order 6. Further, σ2 induces an automorphism of O, and thus ofp^h, that extends to an automorphism of R_h, also calledσ₂, mappinga toa². Let

A0 =hσ2i ≤Aut(R_h).

Unit automorphisms. Multiplication by a unit u ∈ U is an automorphism of the additive group O that normalises p^h. Thus it extends to an automorphism µ_u of R_h that fixes a. Let

A₁ =hµ_u |u∈ Ui ≤Aut(R_h).

Central automorphisms. Viewing R_h as a subgroup of CnT_h−1 allows an element of T_h−1 ∼=p^h−1 to act as an automorphism by conjugation. This action of φ ∈p^h−1 is denoted by ν_φ. Such automorphisms fix T_h and R_h/T_h pointwise. Let

A₂ =hν_φ|φ∈p^h−1i ≤Aut(R_h).

Lemma 5.4 Aut(R_h) = (A₀nA₁)nA₂, and is isomorphic to (Aut(C)nU)nT_h−1. Proof: Since p^h is a characteristic subgroup ofR_h, it follows that Aut(R_h) maps into Aut(C); and this map is onto, since A₀ maps onto Aut(C). It remains to prove that A₁nA₂ is the kernel of this homomorphism. This kernel maps, by restriction, into the group of automorphisms of p^h as C-module; that is to say, as O-module. But p^h is a free O-module of rank 1, so this automorphism group is naturally isomorphic to the group of unitsU ofO, and so the subgroupA₁ shows that this restriction map is onto.

Finally, we need to verify thatA₂ is the kernel of this restriction map. This kernel, as it centralises both R_h/p^h and p^h, consists of the group of derivations of C into the C-module p^h. Now H¹(C, K) = 0, since C has order 9, and 9 is invertible in K. Thus, if R_h = Cnp^h is embedded, in the natural way, into C nK, then every derivation of C into p^h becomes an inner derivation induced by an element ofK. But the inner derivations induced by conjugation by elements of K that normaliseR_h are those induced by conjugation by elements of p^h−1. The result follows. •

5.4 Some number theory

As the descriptions of Aut(R_j−3) and Hom_C(p^j−3∧p^j−3,p^2j−3) exhibit, number theory plays a role in the construction of skeleton groups. We now present some number- theoretic results that help to solve the isomorphism problem for skeleton groups in Sj.

AsO is a local ring with unique maximal idealp, its unit group U =O \p. Recall the structure of U from [12, Chapter 15].

Lemma 5.5 For i >0, let Uⁱ = 1 +pⁱ and κ_i = 1 + (θ−1)ⁱ.

(a) U = U0 > U1 > U2 > . . . is a filtration of U whose quotients are cyclic, and respectively generated by −1, θ, κ₂, θ³ and κ_i for i≥4.

(b) The torsion subgroup of U has order 18, and is generated by θ and −1.

(c) The exponential map defines an isomorphism p⁴ → U⁴.

Recall that a subgroupU of a groupV coversa normal sectionA/B of V ifA/B ≤ U B/B.

Lemma 5.6 Let ρ: U → U :u7→ σ₂(u)σ₋₁(u)u⁻¹. Then ρ(U) is a subgroup of index 3⁴ in U that covers Ui/Ui+1 if and only if i6∈ {1,3,5,11}.

Proof: First we considerρ(U⁴). Letτ :p⁴ →p⁴ be defined byx7→σ₂(x) +σ₋₁(x)−x.

The image τ(p⁴) maps under the exponential map onto ρ(U4), sop⁴/τ(p⁴)∼=U4/ρ(U4).

To determine the image of τ, observe that σ₂ is an automorphism of order 6. Thus it is diagonalisable with eigenvalues wⁱ for i = 0, . . . ,5, where w := −θ³ is a primitive sixth root of unity. As σ₋₁ = σ³₂, it follows that τ is diagonalisable with eigenvalues {wⁱ+w³ⁱ −1|0≤i≤5}. Hence det(τ) =−9 and the image of τ has index 3² inp⁴. Thus ρ(U⁴) has index 3² inU⁴.

Next, we determineρ(U) modulo U12= 1 + 9O. A routine calculation shows:

ρ(−1) = −1;

ρ(θ) = 1;

ρ(κ2) ≡ κ2θ⁶κ4κ6κ7κ8κ9 modU¹²; ρ(θ³) = 1;

ρ(κ₄) ≡ κ₄κ₅κ₆κ₁₀κ₁₁modU12; ρ(κ₅) ≡ κ₆κ²₇κ²₁₀modU¹²; ρ(κ₆) ≡ κ₆κ²₉ modU¹²; ρ(κ₇) ≡ κ²₈κ²₉κ²₁₀κ₁₁ modU¹²; ρ(κ8) ≡ κ8κ²₉κ10κ11 modU¹²; ρ(κ₉) ≡ 1 mod U12;

ρ(κ₁₀) ≡ κ₁₀κ₁₁modU12; ρ(κ₁₁) ≡ 1 mod U¹².

Thusρ(U⁴) covers neitherU⁵/U⁶norU¹¹/U¹². Sinceρ(U⁴) has index 3²inU⁴, it contains

U¹². The result follows. •

Hence U/ρ(U) has order 81 and is generated by the cosets with representatives θ, κ₅, κ₁₁. A routine calculation shows that κ³₅ ≡ κ²₁₁modU¹², so θ and κ₅ suffice.

Defining V := (Z/9Z)², we obtain an isomorphism of abelian groups ϕ:U/ρ(U)→V defined by

θ^u1κ^u₅²ρ(U)7→(u₁, u₂).

Lemma 5.7 The Galois automorphism σ₂ acts on U/ρ(U) as V →V : (u₁, u₂)7→(2u₁,2u₂).

Proof: By definition, σ2(θ) = θ². A routine calculation shows that σ2(κ5)≡κ²₅κ²₆κ²₇κ8κ9κ11 modU¹².

This yields σ2(κ5)≡κ²₅ modρ(U). •

5.5 A solution of the isomorphism problem

We show that orbit isomorphisms solve the isomorphism problem completely for the skeleton groups in S^j.

Lemma 5.8 Letγ andγ⁰ be two surjections inHom_C(p^j−3∧p^j−3,p^2j−3). ThenR_j−3,γ,m andR_j−3,γ⁰_,mare isomorphic if and only if there existsα∈Aut(R_j−3)withα(γ_m) = γ_m⁰ . Proof: By Lemma 3.3, we only have to show that, if R_j−3,γ,m and R_j−3,γ⁰_,m are isomorphic, then there exists an automorphism α of R_j−3 with α(γ_m) = γ_m⁰ . But T_j−3/T_m+j−3 is characteristic inR_j−3,γ,m and R_j−3,γ⁰_,m. Thus, ifR_j−3,γ,m and R_j−3,γ⁰_,m are isomorphic, then an isomorphism between them induces automorphisms α₁ and α₂ of C =hai andT_j−3/T_2j−3 respectively. These automorphisms form a compatible pair;

namely, a^−α1t^α2a^α1 = (a⁻¹ta)^α2 for all t in T_j−3/T_2j−3. Since T_j−3 is a free O-module, α₂ lifts to an automorphism α₃ of R_j−3 such that α₁ and α₃ form a compatible pair, and hence define an automorphism α of R_j−3. Clearlyα satisfiesα(γ_m) = γ_m⁰ . • We must investigate the action of Aut(R_j−3) on the homomorphisms induced by the surjections to solve the isomorphism problem for skeleton groups. Observe that A₂ ≤ Aut(R_j−3) acts trivially on them. Hence it remains to determine the (A₀nA₁)-orbits on the image of Hom_C(p^j−3∧p^j−3,p^2j−3) in Hom_C(p^j−3/p^2j−3∧p^j−3/p^2j−3,p^2j−3/p^m+j−3).

Throughout letn =m−j, soR_j−3,γ,mis a group of depthn inSj. Recall the definition of ϑ from Equation (1).

Lemma 5.9 Let c, c⁰ ∈ U. The surjections cϑ and c⁰ϑ induce the same element of Hom_C(p^j−3/p^2j−3∧p^j−3/p^2j−3,p^2j−3/p^2j−3+n) if and only if c≡c⁰ modUⁿ.

Proof: Let c ∈ Un, so c = 1 +e for e ∈ pⁿ. If x, y ∈ p^j−3, then cϑ(x ∧ y) = ϑ(x∧y) +eϑ(x∧y) and eϑ(x∧y)∈p^2j−3+n. The converse is similar. • By Lemma 5.9, the desired orbits correspond to the (A₀nA₁)-orbits on

Ω_n=U/Un.

We first consider the action of A₁. Using the definition of ρ from Lemma 5.6, for µ_u ∈A₁ and c∈ U

(µ_u(cϑ))(x∧y) = (cϑ(xu⁻¹∧yu⁻¹))u

= c(σ₂(xu⁻¹)σ₋₁(yu⁻¹)−σ₂(yu⁻¹)σ₋₁(xu⁻¹))u

= c(σ₂(x)σ₂(u⁻¹)σ₋₁(y)σ₋₁(u⁻¹)−σ₂(y)σ₂(u⁻¹)σ₋₁(x)σ₋₁(u⁻¹))u

= cσ₂(u⁻¹)σ₋₁(u⁻¹)u(σ₂(x)σ₋₁(y)−σ₂(y)σ₋₁(x))

= cρ(u⁻¹)ϑ(x∧y).

Thus A₁ acts on Ω_n via multiplication by ρ(U). The orbits of this action correspond to the cosets

∆_n:=U/ρ(U)Uⁿ.

Lemma 5.6 shows that ∆_n has at most 3⁴ elements for every n. As A₁ is normal in A₀ nA₁, the orbits under the action of A₁ are blocks for the orbits of A₀ nA₁. It remains to determine the orbits of A1 on the elements of ∆_n. For c∈ U

(σ₂(cϑ))(x∧y)) = σ₂(c(ϑ(σ₂⁻¹(x)∧σ₂⁻¹(y))))

= σ₂(cσ₂⁻¹(ϑ(x∧y)))

= σ2(c)ϑ(x∧y). (2)

Lemma 5.7 shows that σ₂ acts as multiplication by the diagonal matrix 2 0

0 2

on V ∼=U/ρ(U). This allows us to read off the orbits of A₀nA₁ on ∆_n.

Theorem 5.10 The skeleton S^j is isomorphic to the first j −χ_j levels in Figure 1, where j ≥7 and χ_j = 0 if 3 divides j and χ_j = 1 otherwise.

Proof: The root of this tree corresponds to a mainline group, and the nodes at depth n correspond to groups defined by γ = cϑ for c ∈ U/Un, or rather to orbits of such parameters by Lemmas 5.8 and 5.9. Thus the vertex of depth 1 corresponds to c = ±1 modU¹. These two values of c lie in the same orbit, as they are in the same coset modulo ρ(U). The two vertices of depth 2 arise from the parameters c ∈ {1, θ, θ²}modU². These last two are in the same orbit under σ₂. The left vertex of depth 2 corresponds to c = θ, and the right vertex to c = 1. The three vertices of

depth 4 arise from c = θ^k modU⁴ with k determined modulo 9. The leftmost node we take to be defined by θ, and may equally be defined by θ^k for any k prime to 1 modulo 3. The central node we take to be defined by c = θ³, the alternative c = θ⁶ being in the same orbit as this value by the action of σ₂. The rightmost node is defined by c= 1. The eight nodes of depth 6, from left to right, we take to be defined by c = θκ₅, θ, θκ²₅;θ³κ₅, θ³, θ³κ²₅;κ₅,1 modU6. The groups of depth 12 are defined by c=θκ⁴₅, θκ₅, θκ⁷₅;θκ³₅, θ, θκ⁶₅;θκ²₅, θκ⁵₅, θκ⁸₅;θ³κ₅;θ³κ³₅, θ³, θ³κ⁶₅;θ³κ²₅;κ₅;κ³₅,1 modU¹². • Hence the 17 isomorphism types of groups of depth at least 12 in S^j are obtained by using the homomorphismγ =θ^u1κ^u₅²ϑwith the values ofu₁ andu₂ listed in Table 1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

u1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 0 0 0

u₂ 4 1 7 3 0 6 2 5 8 1 3 0 6 2 1 3 0

Table 1: Representative units for the groups in Sj of depth at least 12

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PP PP

Q Q

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HH HH

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r r r r r r r r r r r r r r r r r

r r r r r r r r r r r r r r r r r

Figure 1: The skeleton Sj of Bj

5.6 Presentations for the skeleton groups in B

Lemma 5.11 Let γ =θ^u1κ^u₅²ϑ and let 7≤j ≤m≤2j−χ_j. Let

f(x) = (−1)^j−3x^2j−5(x+ 1)^(u1+2+8j)(x+ 2)^j−3(x²+ 3x+ 3)(x⁵+ 1)^u2 ∈Z[x], and let a_i denote the coefficient of xⁱ in f(x) for 2j−5≤i≤m+j−4. Let α and τ be two abstract group generators and let τ_i = [τ,_iα] for i≥0. Let

r= [τ₁, τ₀](

m+j−4

Y

i=2j−5

τ_i^a₋ⁱ_j+3)⁻¹. Then R_j−3,γ,m has a presentation

{α, τ |α⁹ = (τ α³)³ = [τ, α³, τ] = [τ, τ^α4][τ^α, τ^α3] = [τ, τ^α5][τ^α2, τ^α3] = [τ,_mα] =r = 1}. Proof: Let F denote the free group on {α, τ}, and T the normal closure of {τ} inF. Let Gdenote the group defined by the presentation, and let H =G/L_j−2(G). We use the same notation for elements and subgroups of F, and the images of these elements and subgroups in G and H; the context will resolve ambiguities.

We first check that the relations are satisfied inR_j−3,γ,m, when αand τ stand for a and (θ−1)^j−3 respectively.

Clearlya⁹ = 1, and so the relation (τ α³)³ = 1 reduces tos·s^θ3·s^θ6 = 0, wheres = (θ−1)^j−3, and the twisted operation defined by γ is denoted by ‘·’. Butϑ(u∧uθ³) = 0 for every u ∈ p^j−3, so s·s^θ3 ·s^θ6 = s+s^θ3 +s^θ6 = 0, and the relation [τ, α³, τ] = 1 follows from the same identity. Similarly [τ, τ^α4][τ^α, τ^α3] = [τ, τ^α5][τ^α2, τ^α3] = 1 follows from the identity ϑ(u∧θ⁴u) +ϑ(θu∧θ³u) =ϑ(u∧θ⁵u) +ϑ(θ²u∧θ³u) = 0.

Finally, to check the relationr = 1, we calculate [τ₀, τ₁] = γ((θ−1)^j−3 ∧(θ−1)^j−2)

= θ^u1(1 + (θ−1)⁵)^u2ϑ((θ−1)^j−3∧(θ−1)^j−2)

= θ^u1(1 + (θ−1)⁵)^u2((θ²−1)^j−3(θ⁻¹ −1)^j−2−(θ⁻¹−1)^j−3(θ²−1)^j−2)

= θ^u1(1 + (θ−1)⁵)^u2((θ−1)^j−3(θ+ 1)^j−3(θ−1)^j−2(−1)^j−2θ^−(j−2)

−(θ−1)^j−3θ^−(j−3)(−1)^j−3(θ−1)^j−2(θ+ 1)^j−2)

= θ^u1(1 + (θ−1)⁵)^u2(θ−1)^2j−5(θ+ 1)^j−3(−1)^j−2θ^−(j−2)(θ+ 1 +θ²)

= θ^u1−j+2(1 + (θ−1)⁵)^u2(θ−1)^2j−5(θ+ 1)^j−3(−1)^j−2(θ²+θ+ 1).

Now substituting x forθ−1 gives rise to the polynomial f(x), as required. Note that the two coefficients a_2j−5 and a_2j−4 of f(x) are both multiples of 3, and a_2j−3 is not a multiple of 3, reflecting the fact that γ maps p^j−3∧p^j−3 ontop^2j−3.

We now consider H. Define τ^(k) =τ^αk for k ≥ 0. Observe that T /T⁰ is generated by {τ^(k) : 0 ≤ k ≤ 5}, since the relations α⁹ = (τ α³)³ = 1 imply that T /T⁰ is a homomorphic image of the additive group of integers in the ninth cyclotomic number