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


Academic year: 2022

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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


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/Lc(P), where Lc(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)ps for some s ∈ {0, . . . , r−1}.

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

[Ti, Q] form a chain T = T0 > T1 > . . . > Ti > Ti+1 > . . . > {0} with [Ti : Ti+1] = p and Ti+d = pTi for 0 ≤ i < ∞. We set T := {0}. This chain extends to a doubly infinite series · · · > T2 > T1 > T0 > T1 > T2 >· · · and again Ti+1 has index p in Ti 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/Li+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/Li+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/Li+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/Li+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/Li+1(G). The quotients G/Li+1(G), G/Li+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/Lj+1(G) and all of its descendants that are not descendants of G/Lj+2(G). Thus Bj 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/Lj+1(G) is denoted by Bj,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 πj :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/Li+1(G) into d sequences (Bi+e,Bi+e+d,Bi+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 Bi+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 ofGbyLi(G) fori >0; and a left-normed commutator [a, b, . . . , b

| {z }


] 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 defineTi∧Tj =Tj∧Ti to be theZp-submodule ofTi∧Ti spanned by {s∧t |s ∈Ti, t∈Tj}.

3.1 Twisting homomorphisms

Let γ :T ∧T →T be a ZpQ-module homomorphism. Then γ(T`∧T) is a Q-invariant subgroup of T for every `≥0. Let γ(T ∧T) =Tj for j ≥0, and letγ(Tj∧T) =Tk. If j ≤m ≤k, then γ induces a homomorphism γm :T /Tj∧T /Tj →Tj/Tm defined by

γm(a+Tj ∧b+Tj) =γ(a∧b) +Tm.

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


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

(a+Tm)·(b+Tm) = (a+b+Tm) + 12γm(a+Tj ∧b+Tj).

The resulting groupTγ,m := (T /Tm,·) has order pm. Commutators are evaluated easily in Tγ,m as

[a+Tm, b+Tm] =γm(a+Tj ∧b+Tj).

Ifm =j, then γm is the trivial homomorphism, and Tγ,m is abelian. If j < m≤ k then Tγ,m has derived subgroupTj/Tm and class precisely 2. Also Tγ,n is a quotient of Tγ,m if j ≤n ≤m.

Lemma 3.1 With the above notation, let γ(T ∧T) =Tj and γ(Tj ∧T) =Tk, 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 Tγ, ∼= (T,+).

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

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


(a) This follows directly from the definition.

(b) Writej =id+ewith 0≤e < d. ThenTk=γ(Tj∧T) =γ(piTe∧T) =piγ(Te∧T), and Tj+d = pTj = γ(pT ∧T) ≤ γ(Te ∧T) ≤ γ(T ∧T) = Tj. Hence piTj+d ≤ Tk ≤ piTj 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) =piTj =Tj+id, and

piγ(Tj+id∧T) =p2iγ(Tj ∧T) =p2iTk=Tk+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) =Tj andγ(Tj∧T) =Tk, andj ≤m≤k < ∞. Since the natural action of Q onT /Tm respects the new multiplication induced by γ, we can define a skeleton groupGγ,m :=QnTγ,m. Ifj is sufficiently large, thenGγ,m is a group of depthm−j in the branch of T(G) with rootQnT /Tj. Lemma 3.1(c) shows that the homomorphism piγ for i≥0 defines a skeleton groupGpiγ,m+2id of depthm−j+idin the branch with rootQnT /Tj+id. Thus the sequence of branches with roots QnT /Tj+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) = Sj and γ(Sj ∧S) = Sk, and j ≤ m ≤ k. Now Hγ,m = QnSγ,m is the skeleton group defined by γ and m. Assume that the largest mainline quotient of Hγ,m has class j, so H/Lj+1(H) ∼= Hγ,m/Lj+1(Hγ,m). Assume also that j is large enough so that Lj+1(H)≤G. DefineGγ,m as the full preimage in Hγ,m ofG/Lj+1(H).

Then Gγ,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 groupGαforGis defined as an extension determined by α ∈ HomQ(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 Gα for the pro-p-groupQnS. SinceGα is a skeleton group for QnS and containsGα as an appropriately embedded subgroup, Gα 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 Ti fori≥0 is then characteristic inG, each α∈Aut(G) induces an automorphism of T /Ti. Hence we can define an action of Aut(G) on the set of homomorphisms γm induced by surjections γ ∈ HomQ(T ∧T, Tj). Namely, for x, y ∈T /Tj let

α(γm)(x∧y) := α(γm1(x)∧α1(y))).

Lemma 3.3 Let γ and γ0 be two surjections in HomQ(T ∧T, Tj), and assume that there exists α∈Aut(G) with α(γm) =γm0 . Then Gγ,m∼=Gγ0,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 /Tm defined by γ and γ0 respectively then

α((a+Tmγ(b+Tm)) = α((a+b+Tm) + 12γm(a+Tj∧b+Tj))

= α(a+Tm) +α(b+Tm) + 12α(γm(a+Tj∧b+Tj))

= α(a+Tm) +α(b+Tm) + 12γm0 (α(a+Tj)∧α(b+Tj)))

= α(a+Tmγ0 α(b+Tm).


Thus α induces an isomorphism Tγ,m → Tγ0,m. Since G = QnT, we deduce that Gγ,m =QnTγ,m and the mapGγ,m →Gγ0,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

Hγ,m ∼=Hγ0,m and thusGγ,m ∼=Gγ0,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 Lc(P) be the last non-trivial term of its lower central series. If P/Lc(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 Bj consisting of skeleton groups defines Sj, 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 =zf,[t, ta] =zg, ttata2 =zh, 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= 2·3s 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 +θ36 = 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 kw for the image of k ∈ K under w ∈ W. Note that ka = kθ, and thus a acts as multiplication by θ. Further, kb = σ4(k), where, for i prime to 3, the map σi 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 Ti for the subgroup corresponding topi. If t0 ∈T corresponds to 1∈ O, thenti = [t0,ia]∈Ti\Ti+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) Gi :=ha, bti−1i fori= 1,2,3, so Gi =D·Ti and has index 3 in its minimal split supergroup Hi1 :=ha, b, ti1i.

(c) ha, yti2i and ha, ya1ti2i for i= 2,3,4. Each subgroup is isomorphic to W ·Ti and has index 9 in W nTi−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(Gi) for 1 ≤ i ≤ 3; in Section 6 we prepare for the latter by studying T(Hi) 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 3j+2, and let Sj be the skeleton of Bj.

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 ∈HomC(O ∧ O,pj), and their imagesγm in HomC(O/pj∧ O/pj,pj/pm), 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 Rγ,m := CnTγ,m have order 3m+2, class m, and depth m−j in Sj. We first describe HomC(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 Sj, 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 σi 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)σ1(y)−σ1(x)σ2(y). (1) Lemma 5.1 The map ϑ is an element of HomC(O ∧ O,O). If i and j are non- negative integers, then ϑ maps pi ∧pj onto pi+j+ε, where ε = 3 if i ≡ j mod 3, and ε= 2 otherwise.

Proof: Clearly ϑ(θx∧θy) = σ2(θx)σ1(θy)−σ1(θx)σ2(θy) = θϑ(x∧y). Henceϑ is compatible with the action of θ, and thusϑ∈HomC(O ∧ O,O). The image of pi∧pj


underϑ is generated byϑ((θ−1)iθu1∧(θ−1)jθu2) 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)iθu1 ∧(θ−1)jθu2)

= (θ2−1)j+eθ2(u2+f)1−1)jθu2 −(θ1−1)j+eθ(u2+f)2 −1)jθ2u2

= (θ2−1)j1−1)j(θ−1)eθu2fe[(1 +θ)eθ3f+e−(−1)e]

= (θ−1)i+jc(i, j, u1, u2) [(1 +θ)eθ3f+e−(−1)e],

where c(i, j, u1, u2) is a unit. Consider the term (1 +θ)eθ3f+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) HomC(pi∧pi,pi+j) ={c(θ−1)j−i−3ϑ|c∈ O}=pj−i−3ϑ.

(b) HomC(pi∧pi,pi+j)\HomC(pi∧pi,pi+j+1) = (θ−1)ji3Uϑ.

Proof: Lemma 5.1 shows that (θ −1)ji3ϑ is a surjective element of HomC(pi ∧ pi,pi+j). By [14, Theorem 11.4.1], the set{ϑ}is aK-basis of HomC(K∧K, K). Hence HomC(pi∧pi,pi+j) =pji3ϑ for every j ≥0. Also pl\pl+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)j3 introduces technical complications to these computations. We avoid these by adjusting our notation.

Let Rh = C nTh ∼= C nph for h ≥ 0. Then R0 = R and Rh ∼= R for h ≥ 1.

Using Rj3 instead of R0, the skeleton groups in Bj correspond to the Z3C-module surjections

γ :pj3∧pj3 →p2j3.

Lemma 5.2 shows that these surjections can be written as cϑ for some unit c ∈ U, avoiding (θ−1)j3.

Ifj ≤m ≤k, then the new surjection γ induces the homomorphism γm :pj3/p2j3∧pj3/p2j3 →p2j3/pm+j3.

As with the previous notation, γm defines a multiplication ‘·’ on the set pj3/pm+j3. We denote the resulting group (pj−3/pm+j−3,·) byTj−3,γ,m. It has order 3m and derived subgroup p2j3/pm+j3. NowRj3,γ,m=CnTj3,γ,mis a skeleton group inSj of order 3m+2, class m, and depthm−j.


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

Proof: Lemma 5.1 implies that ϑ maps pj−3∧pj−3 onto p2j−3, andp2j−3∧pj−3 onto p3j6+ε, where ε= 3 if 3 divides j, and ε = 2 otherwise. HenceRj−3,γ,m is defined for

j ≤m ≤2j −3 +ε= 2j−χj. •

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

5.3 The automorphism group of R


We construct the automorphism group of Rh 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 ofph, that extends to an automorphism of Rh, also calledσ2, mappinga toa2. Let

A0 =hσ2i ≤Aut(Rh).

Unit automorphisms. Multiplication by a unit u ∈ U is an automorphism of the additive group O that normalises ph. Thus it extends to an automorphism µu of Rh that fixes a. Let

A1 =hµu |u∈ Ui ≤Aut(Rh).

Central automorphisms. Viewing Rh as a subgroup of CnTh−1 allows an element of Th1 ∼=ph1 to act as an automorphism by conjugation. This action of φ ∈ph1 is denoted by νφ. Such automorphisms fix Th and Rh/Th pointwise. Let

A2 =hνφ|φ∈ph1i ≤Aut(Rh).

Lemma 5.4 Aut(Rh) = (A0nA1)nA2, and is isomorphic to (Aut(C)nU)nTh1. Proof: Since ph is a characteristic subgroup ofRh, it follows that Aut(Rh) 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 ph as C-module; that is to say, as O-module. But ph 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 Rh/ph and ph, consists of the group of derivations of C into the C-module ph. Now H1(C, K) = 0, since C has order 9, and 9 is invertible in K. Thus, if Rh = Cnph is embedded, in the natural way, into C nK, then every derivation of C into ph becomes an inner derivation induced by an element ofK. But the inner derivations induced by conjugation by elements of K that normaliseRh are those induced by conjugation by elements of ph1. The result follows. •


5.4 Some number theory

As the descriptions of Aut(Rj3) and HomC(pj−3∧pj−3,p2j−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 κi = 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 κi 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)σ1(u)u1. 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) +σ1(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 σ1 = 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

θu1κu52ρ(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 σ25)≡κ25κ26κ27κ8κ9κ11 modU12.

This yields σ25)≡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 Sj.

Lemma 5.8 Letγ andγ0 be two surjections inHomC(pj3∧pj3,p2j3). ThenRj−3,γ,m andRj3,γ0,mare isomorphic if and only if there existsα∈Aut(Rj3)withα(γm) = γm0 . Proof: By Lemma 3.3, we only have to show that, if Rj−3,γ,m and Rj−3,γ0,m are isomorphic, then there exists an automorphism α of Rj3 with α(γm) = γm0 . But Tj3/Tm+j3 is characteristic inRj3,γ,m and Rj3,γ0,m. Thus, ifRj3,γ,m and Rj3,γ0,m are isomorphic, then an isomorphism between them induces automorphisms α1 and α2 of C =hai andTj3/T2j3 respectively. These automorphisms form a compatible pair;

namely, a−α1tα2aα1 = (a1ta)α2 for all t in Tj3/T2j3. Since Tj3 is a free O-module, α2 lifts to an automorphism α3 of Rj−3 such that α1 and α3 form a compatible pair, and hence define an automorphism α of Rj3. Clearlyα satisfiesα(γm) = γm0 . • We must investigate the action of Aut(Rj3) on the homomorphisms induced by the surjections to solve the isomorphism problem for skeleton groups. Observe that A2 ≤ Aut(Rj3) acts trivially on them. Hence it remains to determine the (A0nA1)-orbits on the image of HomC(pj−3∧pj−3,p2j−3) in HomC(pj−3/p2j−3∧pj−3/p2j−3,p2j−3/pm+j−3).

Throughout letn =m−j, soRj−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 HomC(pj3/p2j3∧pj3/p2j3,p2j3/p2j3+n) if and only if c≡c0 modUn.


Proof: Let c ∈ Un, so c = 1 +e for e ∈ pn. If x, y ∈ pj3, then cϑ(x ∧ y) = ϑ(x∧y) +eϑ(x∧y) and eϑ(x∧y)∈p2j3+n. The converse is similar. • By Lemma 5.9, the desired orbits correspond to the (A0nA1)-orbits on


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

u(cϑ))(x∧y) = (cϑ(xu1∧yu1))u

= c(σ2(xu11(yu1)−σ2(yu11(xu1))u

= c(σ2(x)σ2(u11(y)σ1(u1)−σ2(y)σ2(u11(x)σ1(u1))u

= cσ2(u11(u1)u(σ2(x)σ1(y)−σ2(y)σ1(x))

= cρ(u1)ϑ(x∧y).

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


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

2(cϑ))(x∧y)) = σ2(c(ϑ(σ21(x)∧σ21(y))))

= σ2(cσ21(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 Sj is isomorphic to the first j −χj levels in Figure 1, where j ≥7 and χj = 0 if 3 divides j and χj = 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 = θk modU4 with k determined modulo 9. The leftmost node we take to be defined by θ, and may equally be defined by θk 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, θκ253κ5, θ3, θ3κ255,1 modU6. The groups of depth 12 are defined by c=θκ45, θκ5, θκ75;θκ35, θ, θκ65;θκ25, θκ55, θκ853κ53κ35, θ3, θ3κ653κ25535,1 modU12. • Hence the 17 isomorphism types of groups of depth at least 12 in Sj are obtained by using the homomorphismγ =θu1κu52ϑwith 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


























@ PP









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 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 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 =θu1κu52ϑ and let 7≤j ≤m≤2j−χj. Let

f(x) = (−1)j3x2j5(x+ 1)(u1+2+8j)(x+ 2)j3(x2+ 3x+ 3)(x5+ 1)u2 ∈Z[x], and let ai 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 τi = [τ,iα] for i≥0. Let

r= [τ1, τ0](




τiaij+3)1. Then Rj3,γ,m has a presentation

{α, τ |α9 = (τ α3)3 = [τ, α3, τ] = [τ, τα4][τα, τα3] = [τ, τα5][τα2, τα3] = [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/Lj2(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 inRj3,γ,m, when αand τ stand for a and (θ−1)j3 respectively.

Clearlya9 = 1, and so the relation (τ α3)3 = 1 reduces tos·sθ3·sθ6 = 0, wheres = (θ−1)j−3, and the twisted operation defined by γ is denoted by ‘·’. Butϑ(u∧uθ3) = 0 for every u ∈ pj3, so s·sθ3 ·sθ6 = s+sθ3 +sθ6 = 0, and the relation [τ, α3, τ] = 1 follows from the same identity. Similarly [τ, τα4][τα, τα3] = [τ, τα5][τα2, τα3] = 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)j3 ∧(θ−1)j2)

= θu1(1 + (θ−1)5)u2ϑ((θ−1)j3∧(θ−1)j2)

= θu1(1 + (θ−1)5)u2((θ2−1)j31 −1)j2−(θ1−1)j32−1)j2)

= θu1(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)

= θu1(1 + (θ−1)5)u2(θ−1)2j5(θ+ 1)j3(−1)j2θ(j2)(θ+ 1 +θ2)

= θu1j+2(1 + (θ−1)5)u2(θ−1)2j5(θ+ 1)j3(−1)j22+θ+ 1).

Now substituting x forθ−1 gives rise to the polynomial f(x), as required. Note that the two coefficients a2j5 and a2j4 of f(x) are both multiples of 3, and a2j3 is not a multiple of 3, reflecting the fact that γ maps pj−3∧pj−3 ontop2j−3.

We now consider H. Define τ(k)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


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