## 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 p^{n} 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 p^{n}, 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 p^{n} 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 T_{0} := T and T_{i+1} :=

[T_{i}, Q] form a chain T = T_{0} > T_{1} > . . . > 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} > T_{0} > T_{1} > T_{2} >· · · 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,_{i}b].

### 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 = T_{0} > T_{1} > . . .. 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_{`}∧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}) + ^{1}_{2}γ_{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_{p}^{i}_{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(p^{i}T_{e}∧T) =p^{i}γ(T_{e}∧T),
and T_{j+d} = pT_{j} = γ(pT ∧T) ≤ γ(T_{e} ∧T) ≤ γ(T ∧T) = T_{j}. Hence p^{i}T_{j+d} ≤
T_{k} ≤ p^{i}T_{j} or, equivalently, id+j +d ≥ k ≥ id+j. As id = j −e, this yields
2j+ (d−e)≥k≥2j−e.

(c) Note that p^{i}γ(T ∧T) =p^{i}T_{j} =T_{j+id}, and

p^{i}γ(T_{j+id}∧T) =p^{2i}γ(T_{j} ∧T) =p^{2i}T_{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
p^{i}γ for i≥0 defines a skeleton groupG_{p}^{i}_{γ,m+2id} of depthm−j+idin the branch with
rootQnT /T_{j+id}. Thus the sequence of branches with roots QnT /T_{j+id}fori= 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_{α}forGis 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 γ^{0} be two surjections in Hom_{Q}(T ∧T, T_{j}), and assume that
there exists α∈Aut(G) with α(γ_{m}) =γ_{m}^{0} . Then G_{γ,m}∼=G_{γ}^{0}_{,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 ·γ^{0} denote the twisted operations on T /T_{m}
defined by γ and γ^{0} respectively then

α((a+T_{m})·^{γ}(b+T_{m})) = α((a+b+T_{m}) + ^{1}_{2}γ_{m}(a+T_{j}∧b+T_{j}))

= α(a+T_{m}) +α(b+T_{m}) + ^{1}_{2}α(γ_{m}(a+T_{j}∧b+T_{j}))

= α(a+T_{m}) +α(b+T_{m}) + ^{1}_{2}γ_{m}^{0} (α(a+T_{j})∧α(b+T_{j})))

= α(a+T_{m})·γ^{0} α(b+T_{m}).

Thus α induces an isomorphism T_{γ,m} → T_{γ}^{0}_{,m}. Since G = QnT, we deduce that
G_{γ,m} =QnT_{γ,m} and the mapG_{γ,m} →G_{γ}^{0}_{,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_{γ}^{0}_{,m} and thusG_{γ,m} ∼=G_{γ}^{0}_{,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|a^{3} =z^{f},[t, t^{a}] =z^{g}, tt^{a}t^{a}^{2} =z^{h}, z^{3} = [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 H^{2}(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 +θ^{3} +θ^{6} = 0.

The ring of integers O of K is a free Z3-module of rank 6 generated by 1, θ, . . . , θ^{5}.
Let W be the group of Z3-linear maps of O generated by the permutations a =
(1, θ, . . . , θ^{8}) andy= (1, θ^{3}, θ^{6}). Then W has order 81 and is isomorphic to the wreath
product of two cyclic groups of order 3. Further, b = (θ, θ^{4}, θ^{7})(θ^{2}, θ^{8}, θ^{5}) = a^{5}y^{2}ay,
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} = σ_{4}(k), where, for i prime to 3, the map σ_{i} is the
Galois automorphism of K defined by θ 7→θ^{i}.

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=p^{0} >p^{1} >p^{2} > . . ..

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 top^{i}. If t_{0} ∈T
corresponds to 1∈ O, thent_{i} = [t_{0},_{i}a]∈T_{i}\T_{i+1} corresponds to (θ−1)^{i} ∈p^{i}\p^{i+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, t_{0}i so R =CnT.

(b) G_{i} :=ha, bt_{i−}_{1}i 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}_{−}_{1}i.

(c) ha, yt_{i}_{−}_{2}i and ha, ya^{−}^{1}t_{i}_{−}_{2}i 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 3^{9}. 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→ θ^{i} where i is
prime to 3. We first define the map

ϑ :O ∧ O → O:x∧y 7→σ_{2}(x)σ_{−}_{1}(y)−σ_{−}_{1}(x)σ_{2}(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 p^{i} ∧p^{j} onto p^{i+j+ε}, where ε = 3 if i ≡ j mod 3, and
ε= 2 otherwise.

Proof: Clearly ϑ(θx∧θy) = σ_{2}(θx)σ_{−}_{1}(θy)−σ_{−}_{1}(θx)σ_{2}(θy) = θϑ(x∧y). Henceϑ is
compatible with the action of θ, and thusϑ∈Hom_{C}(O ∧ O,O). The image of p^{i}∧p^{j}

underϑ is generated byϑ((θ−1)^{i}θ^{u}^{1}∧(θ−1)^{j}θ^{u}^{2}) for 0≤u_{1}, u_{2} ≤5, asp^{i} = (θ−1)^{i}O
and Ois generated by 1, θ, . . . , θ^{5} as aZ3-module. Lete=i−j andf =u_{1}−u_{2}. Then

ϑ((θ−1)^{i}θ^{u}^{1} ∧(θ−1)^{j}θ^{u}^{2})

= (θ^{2}−1)^{j+e}θ^{2(u}^{2}^{+f)}(θ^{−}^{1}−1)^{j}θ^{−}^{u}^{2} −(θ^{−}^{1}−1)^{j+e}θ^{−}^{(u}^{2}^{+f)}(θ^{2} −1)^{j}θ^{2u}^{2}

= (θ^{2}−1)^{j}(θ^{−}^{1}−1)^{j}(θ−1)^{e}θ^{u}^{2}^{−}^{f}^{−}^{e}[(1 +θ)^{e}θ^{3f}^{+e}−(−1)^{e}]

= (θ−1)^{i+j}c(i, j, u_{1}, u_{2}) [(1 +θ)^{e}θ^{3f}^{+e}−(−1)^{e}],

where c(i, j, u_{1}, u_{2}) is a unit. Consider the term (1 +θ)^{e}θ^{3f}^{+e}−(−1)^{e}: if e≡0 mod 3
then it is in p^{3}; if e ≡ 0 mod 3 and f 6≡ 0 mod 3 then it is in p^{3}\p^{4}; if e 6≡ 0 mod 3

then it is in p^{2}\p^{3}. •

Let U denote the unit group of the ring O.

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

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

(b) Hom_{C}(p^{i}∧p^{i},p^{i+j})\Hom_{C}(p^{i}∧p^{i},p^{i+j+1}) = (θ−1)^{j}^{−}^{i}^{−}^{3}Uϑ.

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

Using R_{j}_{−}_{3} instead of R_{0}, 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,m}is 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

_{h}

We construct the automorphism group of R_{h} for h≥0 from three subgroups.

Galois automorphisms. Observe that σ_{2} 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 calledσ_{2}, mappinga toa^{2}. 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

A_{1} =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

A_{2} =hν_{φ}|φ∈p^{h}^{−}^{1}i ≤Aut(R_{h}).

Lemma 5.4 Aut(R_{h}) = (A_{0}nA_{1})nA_{2}, 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 A_{0} maps onto Aut(C). It remains to prove that
A_{1}nA_{2} 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 subgroupA_{1} shows that this restriction map is onto.

Finally, we need to verify thatA_{2} 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 H^{1}(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 U^{i} = 1 +p^{i} 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, κ_{2}, θ^{3} 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 p^{4} → U^{4}.

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→ σ_{2}(u)σ_{−}_{1}(u)u^{−}^{1}. Then ρ(U) is a subgroup of index
3^{4} in U that covers Ui/Ui+1 if and only if i6∈ {1,3,5,11}.

Proof: First we considerρ(U^{4}). Letτ :p^{4} →p^{4} be defined byx7→σ_{2}(x) +σ_{−}_{1}(x)−x.

The image τ(p^{4}) maps under the exponential map onto ρ(U4), sop^{4}/τ(p^{4})∼=U4/ρ(U4).

To determine the image of τ, observe that σ_{2} is an automorphism of order 6. Thus it
is diagonalisable with eigenvalues w^{i} for i = 0, . . . ,5, where w := −θ^{3} is a primitive
sixth root of unity. As σ_{−}_{1} = σ^{3}_{2}, it follows that τ is diagonalisable with eigenvalues
{w^{i}+w^{3i} −1|0≤i≤5}. Hence det(τ) =−9 and the image of τ has index 3^{2} inp^{4}.
Thus ρ(U^{4}) has index 3^{2} inU^{4}.

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

ρ(−1) = −1;

ρ(θ) = 1;

ρ(κ2) ≡ κ2θ^{6}κ4κ6κ7κ8κ9 modU^{12};
ρ(θ^{3}) = 1;

ρ(κ_{4}) ≡ κ_{4}κ_{5}κ_{6}κ_{10}κ_{11}modU12;
ρ(κ_{5}) ≡ κ_{6}κ^{2}_{7}κ^{2}_{10}modU^{12};
ρ(κ_{6}) ≡ κ_{6}κ^{2}_{9} modU^{12};
ρ(κ_{7}) ≡ κ^{2}_{8}κ^{2}_{9}κ^{2}_{10}κ_{11} modU^{12};
ρ(κ8) ≡ κ8κ^{2}_{9}κ10κ11 modU^{12};
ρ(κ_{9}) ≡ 1 mod U12;

ρ(κ_{10}) ≡ κ_{10}κ_{11}modU12;
ρ(κ_{11}) ≡ 1 mod U^{12}.

Thusρ(U^{4}) covers neitherU^{5}/U^{6}norU^{11}/U^{12}. Sinceρ(U^{4}) has index 3^{2}inU^{4}, it contains

U^{12}. The result follows. •

Hence U/ρ(U) has order 81 and is generated by the cosets with representatives
θ, κ_{5}, κ_{11}. A routine calculation shows that κ^{3}_{5} ≡ κ^{2}_{11}modU^{12}, so θ and κ_{5} suffice.

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

θ^{u}^{1}κ^{u}_{5}^{2}ρ(U)7→(u_{1}, u_{2}).

Lemma 5.7 The Galois automorphism σ_{2} acts on U/ρ(U) as
V →V : (u_{1}, u_{2})7→(2u_{1},2u_{2}).

Proof: By definition, σ2(θ) = θ^{2}. A routine calculation shows that
σ2(κ5)≡κ^{2}_{5}κ^{2}_{6}κ^{2}_{7}κ8κ9κ11 modU^{12}.

This yields σ2(κ5)≡κ^{2}_{5} 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 andγ^{0} be two surjections inHom_{C}(p^{j}^{−}^{3}∧p^{j}^{−}^{3},p^{2j}^{−}^{3}). ThenR_{j−}_{3,γ,m}
andR_{j}_{−}_{3,γ}^{0}_{,m}are isomorphic if and only if there existsα∈Aut(R_{j}_{−}_{3})withα(γ_{m}) = γ_{m}^{0} .
Proof: By Lemma 3.3, we only have to show that, if R_{j−}_{3,γ,m} and R_{j−}_{3,γ}^{0}_{,m} are
isomorphic, then there exists an automorphism α of R_{j}_{−}_{3} with α(γ_{m}) = γ_{m}^{0} . But
T_{j}_{−}_{3}/T_{m+j}_{−}_{3} is characteristic inR_{j}_{−}_{3,γ,m} and R_{j}_{−}_{3,γ}^{0}_{,m}. Thus, ifR_{j}_{−}_{3,γ,m} and R_{j}_{−}_{3,γ}^{0}_{,m}
are isomorphic, then an isomorphism between them induces automorphisms α_{1} and α_{2}
of C =hai andT_{j}_{−}_{3}/T_{2j}_{−}_{3} respectively. These automorphisms form a compatible pair;

namely, a^{−α}^{1}t^{α}^{2}a^{α}^{1} = (a^{−}^{1}ta)^{α}^{2} for all t in T_{j}_{−}_{3}/T_{2j}_{−}_{3}. Since T_{j}_{−}_{3} is a free O-module,
α_{2} lifts to an automorphism α_{3} of R_{j−}_{3} such that α_{1} and α_{3} form a compatible pair,
and hence define an automorphism α of R_{j}_{−}_{3}. Clearlyα satisfiesα(γ_{m}) = γ_{m}^{0} . •
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 A_{2} ≤
Aut(R_{j}_{−}_{3}) acts trivially on them. Hence it remains to determine the (A_{0}nA_{1})-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,m}is a group of depthn inSj. Recall the definition
of ϑ from Equation (1).

Lemma 5.9 Let c, c^{0} ∈ U. The surjections cϑ and c^{0}ϑ 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≡c^{0} modU^{n}.

Proof: Let c ∈ Un, so c = 1 +e for e ∈ p^{n}. 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 (A_{0}nA_{1})-orbits on

Ω_{n}=U/Un.

We first consider the action of A_{1}. Using the definition of ρ from Lemma 5.6, for
µ_{u} ∈A_{1} and c∈ U

(µ_{u}(cϑ))(x∧y) = (cϑ(xu^{−}^{1}∧yu^{−}^{1}))u

= c(σ_{2}(xu^{−}^{1})σ_{−}_{1}(yu^{−}^{1})−σ_{2}(yu^{−}^{1})σ_{−}_{1}(xu^{−}^{1}))u

= c(σ_{2}(x)σ_{2}(u^{−}^{1})σ_{−}_{1}(y)σ_{−}_{1}(u^{−}^{1})−σ_{2}(y)σ_{2}(u^{−}^{1})σ_{−}_{1}(x)σ_{−}_{1}(u^{−}^{1}))u

= cσ_{2}(u^{−}^{1})σ_{−}_{1}(u^{−}^{1})u(σ_{2}(x)σ_{−}_{1}(y)−σ_{2}(y)σ_{−}_{1}(x))

= cρ(u^{−}^{1})ϑ(x∧y).

Thus A_{1} acts on Ω_{n} via multiplication by ρ(U). The orbits of this action correspond
to the cosets

∆_{n}:=U/ρ(U)U^{n}.

Lemma 5.6 shows that ∆_{n} has at most 3^{4} elements for every n. As A_{1} is normal in
A_{0} nA_{1}, the orbits under the action of A_{1} are blocks for the orbits of A_{0} nA_{1}. It
remains to determine the orbits of A1 on the elements of ∆_{n}. For c∈ U

(σ_{2}(cϑ))(x∧y)) = σ_{2}(c(ϑ(σ_{2}^{−}^{1}(x)∧σ_{2}^{−}^{1}(y))))

= σ_{2}(cσ_{2}^{−}^{1}(ϑ(x∧y)))

= σ2(c)ϑ(x∧y). (2)

Lemma 5.7 shows that σ_{2} 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 A_{0}nA_{1} 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 modU^{1}. 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, θ^{2}}modU^{2}. These last two are in the same orbit under σ_{2}. 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} modU^{4} 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 = θ^{3}, the alternative c = θ^{6}
being in the same orbit as this value by the action of σ_{2}. 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 = θκ_{5}, θ, θκ^{2}_{5};θ^{3}κ_{5}, θ^{3}, θ^{3}κ^{2}_{5};κ_{5},1 modU6. The groups of depth 12 are defined by
c=θκ^{4}_{5}, θκ_{5}, θκ^{7}_{5};θκ^{3}_{5}, θ, θκ^{6}_{5};θκ^{2}_{5}, θκ^{5}_{5}, θκ^{8}_{5};θ^{3}κ_{5};θ^{3}κ^{3}_{5}, θ^{3}, θ^{3}κ^{6}_{5};θ^{3}κ^{2}_{5};κ_{5};κ^{3}_{5},1 modU^{12}. •
Hence the 17 isomorphism types of groups of depth at least 12 in S^{j} are obtained
by using the homomorphismγ =θ^{u}^{1}κ^{u}_{5}^{2}ϑwith the values ofu_{1} andu_{2} 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

u_{2} 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

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PP PP XXXX

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HH HH

r

r

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Figure 1: The skeleton Sj of Bj

### 5.6 Presentations for the skeleton groups in B

jLemma 5.11 Let γ =θ^{u}^{1}κ^{u}_{5}^{2}ϑ and let 7≤j ≤m≤2j−χ_{j}. Let

f(x) = (−1)^{j}^{−}^{3}x^{2j}^{−}^{5}(x+ 1)^{(u}^{1}^{+2+8j)}(x+ 2)^{j}^{−}^{3}(x^{2}+ 3x+ 3)(x^{5}+ 1)^{u}^{2} ∈Z[x],
and let a_{i} denote the coefficient of x^{i} in f(x) for 2j−5≤i≤m+j−4. Let α and τ
be two abstract group generators and let τ_{i} = [τ,_{i}α] for i≥0. Let

r= [τ_{1}, τ_{0}](

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 |α^{9} = (τ α^{3})^{3} = [τ, α^{3}, τ] = [τ, τ^{α}^{4}][τ^{α}, τ^{α}^{3}] = [τ, τ^{α}^{5}][τ^{α}^{2}, τ^{α}^{3}] = [τ,_{m}α] =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.

Clearlya^{9} = 1, and so the relation (τ α^{3})^{3} = 1 reduces tos·s^{θ}^{3}·s^{θ}^{6} = 0, wheres =
(θ−1)^{j−}^{3}, and the twisted operation defined by γ is denoted by ‘·’. Butϑ(u∧uθ^{3}) = 0
for every u ∈ p^{j}^{−}^{3}, so s·s^{θ}^{3} ·s^{θ}^{6} = s+s^{θ}^{3} +s^{θ}^{6} = 0, and the relation [τ, α^{3}, τ] = 1
follows from the same identity. Similarly [τ, τ^{α}^{4}][τ^{α}, τ^{α}^{3}] = [τ, τ^{α}^{5}][τ^{α}^{2}, τ^{α}^{3}] = 1 follows
from the identity ϑ(u∧θ^{4}u) +ϑ(θu∧θ^{3}u) =ϑ(u∧θ^{5}u) +ϑ(θ^{2}u∧θ^{3}u) = 0.

Finally, to check the relationr = 1, we calculate
[τ_{0}, τ_{1}] = γ((θ−1)^{j}^{−}^{3} ∧(θ−1)^{j}^{−}^{2})

= θ^{u}^{1}(1 + (θ−1)^{5})^{u}^{2}ϑ((θ−1)^{j}^{−}^{3}∧(θ−1)^{j}^{−}^{2})

= θ^{u}^{1}(1 + (θ−1)^{5})^{u}^{2}((θ^{2}−1)^{j}^{−}^{3}(θ^{−}^{1} −1)^{j}^{−}^{2}−(θ^{−}^{1}−1)^{j}^{−}^{3}(θ^{2}−1)^{j}^{−}^{2})

= θ^{u}^{1}(1 + (θ−1)^{5})^{u}^{2}((θ−1)^{j−}^{3}(θ+ 1)^{j−}^{3}(θ−1)^{j−}^{2}(−1)^{j−}^{2}θ^{−}^{(j−}^{2)}

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

= θ^{u}^{1}(1 + (θ−1)^{5})^{u}^{2}(θ−1)^{2j}^{−}^{5}(θ+ 1)^{j}^{−}^{3}(−1)^{j}^{−}^{2}θ^{−}^{(j}^{−}^{2)}(θ+ 1 +θ^{2})

= θ^{u}^{1}^{−}^{j+2}(1 + (θ−1)^{5})^{u}^{2}(θ−1)^{2j}^{−}^{5}(θ+ 1)^{j}^{−}^{3}(−1)^{j}^{−}^{2}(θ^{2}+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^{α}^{k} for k ≥ 0. Observe that T /T^{0} is generated
by {τ^{(k)} : 0 ≤ k ≤ 5}, since the relations α^{9} = (τ α^{3})^{3} = 1 imply that T /T^{0} is a
homomorphic image of the additive group of integers in the ninth cyclotomic number