# Finite p-groups of maximal class with ‘large’ automorphism groups

Heiko Dietrich and Bettina Eick
From the journal Journal of Group Theory

## Abstract

The classification of p-groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the coclass graph 𝒢 associated with the p-groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph 𝒢 of 𝒢 associated with those p-groups of maximal class whose automorphism group orders are divisible by p-1. We describe the broad structure of 𝒢 by determining its so-called skeleton. We investigate the smallest interesting case p=7 in more detail using computational tools, and propose an explicit version of Conjecture W for 𝒢 for arbitrary p7. Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.

### 1 Introduction

The investigation of the p-groups of maximal class was initiated by Blackburn [1], and has had a long history since then; we refer to the book of Leedham-Green and McKay [11] for details and references. A central tool is the coclass graph𝒢 associated with the p-groups of maximal class: the vertices of 𝒢 are identified with isomorphism type representatives of the considered groups, and there is an edge GH if and only if H/γ(H)G, where γ(H) is the last non-trivial term of the lower central series of H. Investigating the structure of 𝒢 is an approach towards a detailed understanding (and thus towards a possible classification) of the associated groups.

It is well known that 𝒢 consists of an isolated vertex corresponding to the cyclic group of order p2, and an infinite coclass tree whose root is an elementary abelian group of order p2. This tree has a single infinite path starting at its root; the groups on this infinite path S2S3 satisfy |Sn|=pn and are the lower central series quotients of the (unique) infinite pro-p-group of maximal class. Proofs of these facts and further background information can also be found in [11].

We need some graph-theoretic notation to describe 𝒢 in more detail. If there is a path of length k from a group G to a group H in 𝒢, then H is a (k-step) descendant of G, and G is the (k-step) parent of H; a 1-step descendant is an immediate descendant of G. For n the branchn of 𝒢 is the subtree generated by all descendants of Sn which are not descendants of Sn+1. Note that every branch n is a finite tree with root Sn, and the whole coclass tree of 𝒢 is partitioned into its branches, which are connected via the infinite path, see Figure 1. In conclusion, the structure of 𝒢 is determined by the structure of its branches.

### Figure 1

The coclass graph 𝒢 with its coclass tree and branches n.

The depth of a vertex in a rooted tree is its distance from the root; the depth of a rooted tree is the maximal depth of a vertex. For k the pruned branch n(k) is the subtree of n generated by all groups of depth at most k in n. It was proved independently by du Sautoy [18] and Eick and Leedham-Green [8] that for each k there exists l(k)>0 such that n+p-1(k)n(k) for all nl(k). Thus the pruned branches eventually repeat periodically; we call this the first periodicity.

If p{2,3}, then all branches in 𝒢 have depth 1 and therefore can be described by the first periodicity. For larger p, however, the depth of the branches lies between n-m and n+m-3 for m=2p-8, see for example [4, Theorem 1.2]. Hence, the first periodicity is not capable of describing 𝒢 completely. It remains to understand how the branches grow beyond their pruned versions. The work of Leedham-Green and McKay (see [10, 11] and the references there) sheds some light on this. We call a group in 𝒢capable if it is not a leaf, and we define the skeleton𝒮n as the subtree of n generated by all capable groups at depth at most n-m in n. Leedham-Green and McKay [10] introduced a construction for certain skeleton groups and they showed how the isomorphism problem for these groups is related to number theory over the p-adic rational numbers; we briefly recall and then extend this in Section 4.

We say that 𝒢 has finite width if the number of groups of fixed depth in n is bounded by a constant independent of n. It is known that 𝒢 has finite width if and only if p5. This indicates that p=5 is a special case. Indeed, in this case the branches are small enough to be accessible to computer investigations. Moreover, the associated groups are significantly easier to study theoretically than the groups for larger primes. Further references and a more detailed description of explicit results are given in Appendix A.

The situation changes considerably for p7. Here the graph 𝒢 has infinite width and its branches are too complex for a detailed (computational) investigation. In particular, it is also an open question whether or not the groups of maximal class can be classified by an investigation of 𝒢. If such a classification is at all possible, then Coclass Conjecture W (see [9]) proposes an approach: it suggests that there exists an integer k such that for each nk the branch n+p-1 can be constructed from n using two types of periodic patterns. One of these patterns is the first periodicity, the other could be called a second periodicity. We note that the description of the second periodicity in Conjecture W is rather vague and not as explicit as the first periodicity; this may be part of the reason why it is so difficult to investigate. We emphasise that there is only very little evidence for Conjecture W so far and that all the available evidence is in coclass trees of finite width. We exhibit further details on Conjecture W in Appendix A.

#### 1.1 Main results

We consider an arbitrary prime p7, which is fixed throughout this paper, and define constants

d=p-1,=p-32,m=2p-8.

We use the notation 𝒢, n, 𝒮n defined above. The aim of this paper is to study the subgraph 𝒢 of 𝒢 generated by all groups whose automorphism group order is divisible by d. More precisely, we investigate 𝒢 with a view towards understanding the periodicities proposed by Conjecture W; if there is any chance to prove Conjecture W at all, then it seems useful to understand first the possible periodicities in more detail. We denote by n and 𝒮n the subgraphs of n and 𝒮n contained in 𝒢; these are both subtrees of n. Our first result is a complete determination of the skeletons 𝒮n; see Section 6 for its proof.

Theorem 1.1

Let nmax{m,8}.

1. The skeleton 𝒮n has groups at depth 1 ; we denote these by Gn,1,,Gn,.

2. Let H be a descendant of Gn,i at depth e<n-m in 𝒮n. Then H has p immediate descendants in 𝒮n if and only if (emodd){2,4,,d-2}{d-2i}; otherwise, H has one immediate descendant in 𝒮n.

Theorem 1.1 exhibits that each skeleton 𝒮n consists of subtrees starting at depth 1 and each subtree has a well-described branching pattern depending on the parameter i of the root of the subtree. The next corollary is an immediate consequence.

Corollary 1.2

Let nmax{m,8} and e<n-m. The number of groups at depth e in Sn is at least pe/d(-1); in particular, G has infinite width.

We have determined n for p=7 and 10n18 by computer, cf. Section 1.2. Based on this, we formulate a conjectural description for n for arbitrary p7. To describe this conjecture, we define the twig𝒲G of a group G in 𝒮n: this is the subtree of n with root G containing all descendants of G which are not descendants of any proper descendant of G in 𝒮n. Thus each group of n is contained in exactly one twig and the twigs of n are connected by the skeleton of n. We continue to denote the groups of depth 1 in a skeleton 𝒮n by Gn,1,,Gn, and we assume that these are sorted via Theorem 1.1 (b). Note that every group G of depth at least 1 in 𝒮n has exactly one of the groups Gn,i as parent.

Conjecture 1.3

There exists lm such that for all nl and for each G of depth e at least 1 in 𝒮n with parent Gn,i the following holds.

1. If G is not a leaf in 𝒮n, then the isomorphism type of the graph 𝒲G depends on the index i, on emodd, and on nmodd only.

2. If G is a leaf in 𝒮n, then there exists a group G¯ with parent Gn-d,i at depth e-d in 𝒮n-d with 𝒲G𝒲G¯.

Conjecture 1.3 suggests that there are three types of twigs: the twigs of the leaves in 𝒮n, the twigs of the roots of 𝒮n, and the twigs of the other skeleton groups. Note that, by definition, 𝒲G has depth at most 1 for every group G in 𝒮n which is not a leaf. We choose l large enough such that the first periodicity holds for all groups of depth 1 in n; then the twigs of the roots in 𝒮n behave periodically for all nl and thus there are at most l+d different twigs of roots in 𝒢. Conjecture 1.3 suggests that for nl there are only d2 different twigs in 𝒮n for groups that are neither roots nor leaves. The twigs of the leaves of 𝒮n are trees of depth at most 2m+3. Conjecture 1.3 suggests that there are finitely many different twigs of skeleton-leaves; more precisely, if the skeleton 𝒮n has wn leaves, then Conjecture 1.3 proposes that there are at most wl+1++wl+d different twigs arising for the leaves in all 𝒮n with nl.

#### 1.2 Computations for p=7

We have computed some branches for p=7 (and partial branches for p=11) using the computer algebra system GAP [20] and the GAP package AnuPQ which is based on [16]. Figure 2 illustrates Conjecture 1.3 with the branches 10,,16 for p=7. The black parts of the graphs and the bold numbers on the left of vertices describe the skeletons: a number k on the left of a vertex indicates that this vertex and all of its descendants appear k times with the same parent. A number w on the right of a vertex says that this vertex has w immediate descendants in addition to the displayed descendants. The grey parts of the graphs are twigs of the leaves in 𝒮n.

### Figure 2

The branches n (and skeleton 𝒮n) for p=7 and n=10,,16.

#### 1.3 Structure of the paper

In Section 2 we recall some p-adic number theory; these results are important for defining the groups in the skeleton and for solving their isomorphism problem. In Section 4 we recall the construction of the skeleton groups in 𝒮n, and we consider their isomorphism problem and automorphism groups. In Section 5 we then investigate the skeleton groups in 𝒮n in more detail. In particular, we show how they can be constructed up to isomorphism. In Section 6, we prove Theorem 1.1. Appendix A contains a short survey on known periodicity results for coclass graphs.

### 2 Some number theory

Throughout the paper, p and p denote the field of p-adic rational numbers and ring of p-adic integers, respectively. The p-th cyclotomic polynomial

1+x++xp-1p[x]

is irreducible; we consider a fixed root θ and define K=p(θ), so that K is a field extension of degree d=p-1 over p, with p-basis {1,θ,,θd-1}. For a with pa the field automorphism σa:KK is defined by σa(θ)=θa. The Galois group of K is Gal(K)={σa1ad}; it is cyclic and we fix a generator σ=σk. The equation order 𝒪=p[θ] is the maximal order of K; it has {1,θ,,θd-1} as p-basis, and a unique maximal ideal 𝔭=(θ-1). We abbreviate κ=θ-1, so that (κm)=𝔭m for m; this defines a series of ideals through 𝒪. For z and n we denote by (𝔭z)n the direct sum of n copies of 𝔭z (and not the ideal 𝔭nz). For each non-zero wK there exist unique z and a unit up[θ] such that w=κzu; we call val(w)=z the valuation of w. Note that if v,wK are non-zero, then val(vw)=val(v)+val(w). We extend this definition to non-zero n-tuples 𝒗Kn, so that val(𝒗)=z if and only if 𝒗(𝔭z)n(𝔭z+1)n.

#### 2.1 Eigenvalues

The generator σ of the Galois group of K can be considered as a p-linear map of K. The next lemma determines its eigenvalues; it is proved in [10, Lemma 2.3] for z0, but the same proof holds for all z.

Lemma 2.1

The eigenvalues of the Qp-linear map σ:KK are ω0,,ωd-1 and each eigenspace has dimension 1. If wK with val(w)=z is an eigenvector of σ, then the corresponding eigenvalue is ωz. For every zZ there exists an eigenvector w of σ with val(w)=z.

#### 2.2 The group of units

Let 𝒰 be the unit group of 𝒪. For i2 define 𝒰i=1+𝔭i, and let ω be a primitive d-th root of unity in p; we assume throughout that ω is chosen such that ωkmodp where k is defined by the fixed generator σ=σk of Gal(K). It is shown in [13, Satz II.5.3] that

𝒰=ω×θ×𝒰2.

The unit group of p is p𝒰, that is, p=ω×(1+pp). In the course of the paper we will need various maps based on these unit groups. The following lemma investigates one of them.

Lemma 2.2

The map χ:UU, uuσ(u)-1, is a group homomorphism with kerχ=Zp* and U=kerχ×imχ.

#### Proof.

Since σ generates Gal(K), the fixed points of σ in K are exactly the elements of the subfield p of K, hence kerχ=𝒰p=p. Next, we consider the restriction of χ to 𝒰2=1+𝔭2. As shown in [13, Satz II.5.5 and p. 146], the additive group of 𝔭2 is isomorphic to the multiplicative group 𝒰2 via

exp:(𝔭2,+)(𝒰2,),xk=0xkk!.

As this exponential map is compatible with the action of σ, we can translate χ to a map ψ:𝔭2𝔭2, xx-σ(x). Since σ is a diagonalisable p-linear map on 𝔭2 with eigenvalues ω0,ω1,,ωd-1, the map ψ is diagonalisable with eigenvalues 0,1-ω1,,1-ωd-1. Now note that if i{1,,d-1}, then 1-ωi0modp, so 1-ωip; moreover, Lemma 2.1 implies that there exists an eigenvector of ψ in 𝔭2𝔭3. In conclusion, we have shown that

𝔭2=kerψimψ,

and hence 𝒰2 is the direct product of kerχ and imχ restricted to 𝒰2. Finally, note that χ(ω)=1 and χ(θ)=θ1-k. Thus the result follows from the decomposition of 𝒰 as 𝒰=ω×θ×𝒰2. ∎

### 3 The infinite pro p-group of maximal class

The following lemma shows how the structure of K relates to the infinite pro-p-group of maximal class, see [11, Proposition 8.3.2]. From now on, we denote by T=(𝒪,+) the additive group of the ring 𝒪, and let P be the cyclic group of order p generated by θ. We let P act on T by multiplication.

Lemma 3.1

The semidirect product S=TP is an infinite pro-p-group of coclass 1.

The unique maximal S-invariant series through T is T=T1>T2>, where each Ti=(𝔭i-1,+). Moreover, if i2, then Ti=γi(S) is the i-th term in the lower central series of S. For i define Si=S/γi(S), so that S2S3 is the unique maximal infinite path in 𝒢. The automorphism groups of S and its quotients Si are described in the following lemma from [4, Section 4.2]; it implies that the maximal path S2S3 is contained in 𝒢.

Lemma 3.2

Writing elements of S as tuples (t,θi) with tT and iZ, the following hold:

1. The natural restriction π:Aut(S)Aut(T), αα|T, satisfies

imπ𝒰Gal(K)𝑎𝑛𝑑kerπZ1(P,T).

A preimage of (u,σb)𝒰Gal(K) under π is given by

α(u,b):SS,(t,θi)(uσb(t),θib).

The kernel of π is generated by {α(1),α(θ),,α(θd-1)}, where for sT we define

α(s):SS,(t,θi)(t+(1+θ++θi-1)s,θi).
2. If i4, then the natural projection Aut(S)Aut(Si) is surjective and |Out(Si)|=pi-2d2.

### 4 The skeleton groups in 𝒮n

In this section we describe the construction of skeleton groups and their isomorphism problem, based on results of Leedham-Green and McKay [10], see also [11, Section 8.2]. A key ingredient in that construction is homomorphisms from the exterior square TT: this is the pP-module generated by st with s,tT such that for all s,s,tT and zp the following holds: ss=0, hence st=-(ts), and z(st)=(zs)t=s(zt), and (st)+(st)=(s+s)t. The group P acts diagonally on TT, which defines the pP action on TT.

In the following let nmax{m,8} and e{0,,n-m}. Every surjective homomorphism fHomP(TT,Tn) defines an associative multiplication on T/Tn+e via

(t+Tn+e)(s+Tn+e)=t+s+12f(ts)+Tn+e.

We denote the resulting group by Mn,e(f). It is not difficult to show that Mn,e(f) has class 2 and derived subgroup Mn,e(f)=Tn/Tn+e. Since f is a P-module homomorphism, the multiplication in Mn,e(f) is compatible with the action of P, and we can define the group

Cn,e(f)=Mn,e(f)P;

these groups are called constructible in [11]. Each group Cn,e(f) is an extension of the natural Sn-module Tn/Tn+e by the group Sn on the infinite path of 𝒢; in particular, it is a group of depth e in the skeleton 𝒮n. By [4, Theorem 1.3], the groups Cn,e(f) are exactly the groups in the skeleton 𝒮n.

#### 4.1 The structure of HomP⁢(T∧T,T)

The structure of HomP(TT,T) has been investigated by Leedham-Green and McKay. We recall some of the results here for completeness, as we need them in later applications. We refer to [11, Sections 8.2 and 8.3] for further details.

First, TT=FZ, where F is a pP-module of rank =(p-3)/2 generated by κiκi-1 with i=1,,, and Z is a free p-module of rank 1. Let δij denote the Kronecker-delta and recall that σa is an element of Gal(K) for pa. For i{1,,} we define 𝕋iHomP(TT,T) via

𝕋i(κjκj-1)=δijand𝕋i(z)=0 for zZ.

For 2a+1 we define 𝕊aHomP(TT,T) via

𝕊a(xy)=σa(x)σ1-a(y)-σ1-a(x)σa(y).

For an -tuple 𝒄=(c1,,c)K let

𝕊(𝒄)=c1𝕊2++c𝕊+1and𝕋(𝒄)=c1𝕋1++c𝕋.

For a{2,,+2} and i{1,,} let

ba,i=(θa-θ1-a)((θa-1)(θ1-a-1))i-1,

and define the ×-matrix B over K as

B=(b2,1b2,2b2,b3,1b3,2b3,b+1,1b+1,2b+1,)K×.

The next lemma is proved in [11, Theorems 8.3.1 and 8.3.7] and the proof of [11, Proposition 8.3.8].

Lemma 4.1

Let fHomP(TT,T).

1. There exists a unique 𝒄𝒪 with f=𝕋(𝒄).

2. There exists a unique 𝒅K with f=𝕊(𝒅).

3. The matrix B describes a base change from {𝕊2,,𝕊+1} to {𝕋1,,𝕋}, that is, 𝒄=𝒅B.

If 𝒄𝒪, then both 𝕋(𝒄) and 𝕊(𝒄) lie in HomP(TT,T). By Lemma 4.1, {𝕋1,,𝕋} forms an 𝒪-basis for HomP(TT,T). There exists 𝒄K𝒪 with 𝕊(𝒄)HomP(TT,T), which shows that the set {𝕊2,,𝕊+1} generates HomP(TT,T), but not as an 𝒪-module. Nonetheless, the latter generating set plays an important role in the solution of the isomorphism problem, see Section 4.2.

The groups at depth e in 𝒮n can be obtained as Cn,e(𝕊(𝒄)), where 𝕊(𝒄) has image Tn; thus we define

Γn=(𝔭n-1)B-1,
Δn={𝒄KCn,e(𝕊(𝒄))𝒮n}
=ΓnΓn+1,
Δn,e*={𝒄KCn,e(𝕊(𝒄))𝒮n}Δn.

Note that Δn=κΔn-1.

For the next lemma, recall the definition of the valuation given in Section 2.

Lemma 4.2

For i=1,, let 𝐭𝐢 be the i-th row of B-1.

1. We have 𝕋i=𝕊(𝒕𝒊) and val(𝒕𝒊)=-2i+1.

2. If 𝒗=a1𝒕𝟏++a𝒕Δ1 with each aiK, then

val(𝒗)=min{val(ai𝒕𝒊)i=1,,}.

#### Proof.

(a) The proof of [11, Proposition 8.3.8] shows that det(B)𝔭2𝔭2+1, hence B is invertible over K. By Lemma 4.1, the i-th row 𝒕𝒊 of B-1 satisfies 𝕋i=𝕊(𝒕𝒊); it remains to analyse the valuation of 𝒕𝒊. It is straightforward to see that each entry ba,i of B has valuation val(ba,i)=𝔭2i-1, hence

B=Udiag(κ,κ3,,κ2-1),

where U is an × matrix with entries in 𝒰 only. Since det(B)𝔭2𝔭2+1, this implies that det(U)𝒰. Moreover,

B-1=diag(κ-1,κ-3,,κ-2+1)U-1.

Since det(U)𝒰, Cramer’s rule for matrix inverses shows that each entry of U-1 lies in 𝒪. Since det(U-1)𝒰, each row and column of U-1 contains at least one element in 𝒪𝔭. In conclusion, the valuation of the i-th row of U-1 is 0, hence the valuation of the i-th row of B-1 is -2i+1.

(b) Let μ=min{val(ai𝒕𝒊)i=1,,}, J={i1,,val(ai𝒕𝒊)=μ}. Clearly, val(𝒗)μ, and we have to show equality. If |J|=1, then val(𝒗)=μ follows readily; thus we suppose J={i1,,in} with i1<<in and 1<n. Note that val(𝒗)=μ if and only if

(4.1)val(ai1𝒕𝒊𝟏++ain𝒕𝒊𝒏)=μ.

Since val(aij𝒕𝒊𝒋)=μ for all j, we can assume that μ=0: simply replace 𝒗 by a multiple κ-val(𝒗)𝒗. By part (a), each val(𝒕𝒊𝒋)=-2ij+1, so val(aij𝒕𝒊𝒋)=0 implies val(aij)=2ij-1, that is, there is a uniquely defined cij{1,,p-1} such that

aij𝒕𝒊cijκ2ij-1𝒕𝒊mod(𝔭).

Suppose, for a contradiction, that (4.1) is false, that is,

val(ai1𝒕𝒊𝟏++ain𝒕𝒊𝒏)>0.

This means that ai1𝒕𝒊𝟏++ain𝒕𝒊𝒏(𝔭), and so

ci1κ2i1-1𝒕𝒊𝟏++cinκ2in-1𝒕𝒊𝒏(𝔭).

In other words, if (4.1) is false, then there are c1,,c{0,,p-1}, not all 0, such that

(4.2)c1κ1𝒕𝟏+c2κ3𝒕𝟐++cκ2-1𝒕(𝔭).

We show that this is not possible; then (4.1) must be true, and then so is the claim of the lemma.

Since the rows of B-1 are 𝒕𝟏,,𝒕, equation (4.2) is false if and only if the rows of the matrix

M=diag(κ,κ3,,κ2-1)B-1𝒪×

are linearly independent over 𝒪/𝔭, which is isomorphic to the field with p-elements. This is the case if and only if M is invertible over 𝒪/𝔭, if and only if

M-1=Bdiag(κ-1,κ-3,,κ-2+1)𝒪×

is invertible over 𝒪/𝔭. It follows from the definition of the entries ba,i of B that

B=diag(x2,,x+1)V(z2,,z+1)diag(κ1,κ3,,κ2-1),

where

xa=θ1-a(1+θ++θ2a-2),za=(1+θ++θa-1)(1+θ++θ-a)for all a,

and V(z2,,z+1) is a Vandermonde matrix with parameters z2,,z+1. This yields

M-1=diag(x2,,x+1)V(z2,,z+1).

Note that each xa(2a-1modp)mod𝔭0, and all za(-a2+amodp)mod𝔭0 are pairwise distinct. This proves that the determinant of M-1 is a unit in 𝒪/𝔭; therefore M is invertible over 𝒪/𝔭, which proves that (4.2) cannot be true. Thus (4.1) must hold, and the lemma is proved. ∎

#### 4.2 The isomorphism problem and automorphism groups

Recall that the groups in 𝒮n can be constructed as C=Cn,e(𝕊(𝒄)) with 𝒄Δn. By definition, C is an extension of Tn/Tn+e by Sn, and Tn/Tn+e is a fully invariant subgroup of C. Hence the isomorphism problem and the determination of Aut(C) can be approached using the general ideas for group extensions, see for example [17].

We investigate under which conditions two elements of Δn define isomorphic groups. For this undertaking, the group homomorphisms

ρa:𝒰𝒰,uu-1σa(u)σ1-a(u),

with a{2,,+1} play an important role. We first recall an action on Δn, motivated by [10].

Lemma 4.3

The element (u,σb)UGal(K) acts on 𝐜=(c1,,c)Δn via

(u,σb)(𝒄)=(ρ2(u)-1σb(c1),,ρ+1(u)-1σb(c))Δn.

This induces an action of UGal(K) on Γn and on the set of cosets Γn/Γn+e for each eN.

#### Proof.

We explain the origin of this action. Every automorphism βAut(S) acts on fHomP(TT,Tn) via fβ[f]=βf(β-1β-1)|TT; if f is surjective, then so is β[f]. For the following, recall the notation of Lemma 3.2. If β is in the kernel of the map π:Aut(S)Aut(T), then β[f]=f for every fHomP(TT,Tn). Now let β=α(u,b) with (u,σb)𝒰Gal(K). If a{2,,+1}, cK, and xyTT, then a short computation shows that β[c𝕊a](xy)=ρa(u-1)σb(c)𝕊a(xy). This implies that f=𝕊(𝒄) is mapped to β[f]=𝕊((u,σb)(𝒄)), and (u,σb)(𝒄)Δn as required. ∎

Lemma 4.3 allows us to formulate a solution to the isomorphism problem for skeleton groups and a description for their automorphism groups. For this purpose, let C=Cn,e(𝕊(𝒄)) be defined as above and let

λ:Aut(S)Aut(Sn)×Aut(Tn/Tn+e),
λC:Aut(C)Aut(Sn)×Aut(Tn/Tn+e)

be induced by the natural restrictions. It is easy to show that the kernel of λC is isomorphic to Z1(Sn,Tn/Tn+e); its image is described in [17] using cohomology. Here we describe the image of λC in a different way that will be more useful in our setting. Recall from Lemma 3.2 that Aut(S) is an extension of Z1(P,T) by 𝒰Gal(K).

Theorem 4.4

Let nmax{m,8} and e{0,,n-m}; let 𝐜,𝐝Δn.

1. The groups Cn,e(𝕊(𝒄)) and Cn,e(𝕊(𝒅)) are isomorphic if and only if 𝒄+Γn+e and 𝒅+Γn+e lie in the same orbit under the action of 𝒰Gal(K) as defined in Lemma 4.3.

2. The group Aut(Cn,e(𝒄)) is an extension of the p-group Z1(Sn,Tn/Tn+e) by λ(Z1(P,T).Σ), where Σ is the stabiliser of 𝒄+Γn+e in 𝒰Gal(K).

#### Proof.

This follows from results in [4, 3], based on the general cohomological approach outlined in [17]. We briefly summarise the approach since this explains the reduction to the action of 𝒰Gal(K).

(a) Every group of depth e in n is an extension of Tn/Tn+e by the root Sn of n, where Tn/Tn+e carries the obvious Sn-module structure, see [4, Theorem 3.1]. Such extensions can be described by elements of the second cohomology group H2(Sn,Tn/Tn+e), and the isomorphism problem of such extensions can be solved by considering the action of the group of compatible pairs Comp(Sn,Tn/Tn+e), which consists of pairs of compatible automorphisms of Sn and Tn/Tn+e, respectively, see [4, Section 7.1]. By [4, Theorem 7.1], the isomorphism problem for skeleton groups can be solved by considering compatible pairs λ(α)=(α|Sn,α|Tn/Tn+e) defined by αAut(S), acting on certain cohomology classes defined by surjective homomorphisms TTTn. This action is discussed in detail in [3, Section 4.2], and it turns out that one has to consider exactly the automorphisms α(u,b) of S which are defined by elements (u,σb)𝒰Gal(K); the automorphisms α(s) with sT act trivially. Putting all this together, the statement of the theorem follows. We note that in [4, 3] groups of depth e in n are described as extensions of T/Te+1 by Sn; however, Tn/Tn+eT/Te+1 as Sn-modules, and applying a suitable Sn-module isomorphism translates the results to our set-up, cf. [3, Remark 1].

(b) The claim follows from known results about automorphism groups of extensions, together with [4, Lemma 5.4]. We use the notation introduced in part (a) and let γH2(Sn,Tn/Tn+e) be a cohomology class defining Cn,e(𝕊(𝒄)). It is shown in [17] that the image of λC is isomorphic to the stabiliser of γ in Comp(Sn,Tn/Tn+e), and that the kernel of λC is isomorphic to the p-group Z1(Sn,Tn/Tn+e). The proof of [3, Lemma 5.4] now shows that the stabiliser of γ in the group of compatible pairs is isomorphic to the stabiliser of 𝒄+Γn+e in Aut(Sn+e), where αAut(Sn+e) acts as the compatible pair λ(α). Note that Z1(P,T).Σ is the stabiliser of 𝒄+Γn+e in Aut(S); since Aut(S)Aut(Sn+e) is surjective (and its kernel acts trivially on 𝒄+Γn+e), the claim follows. ∎

Leedham-Green and McKay [10] also considered the isomorphism problem using a different approach: they considered the homomorphism defined by commutation in a skeleton group, and investigated how this homomorphism changes when one modifies certain generators of the group. Their results [10, Propositions 1.1 and 1.2] are in line with ours and might be used to prove one direction of the isomorphism problem.

### 5 The skeleton groups in 𝒮n∗

The automorphism group of a skeleton group C=Cn,e(𝒄) with 𝒄Δn is described in Theorem 4.4, and we have

Aut(C)Z1(Sn,Tn/Tn+e).λ(Z1(P,T).Σ),

where Σ is the stabiliser of 𝒄+Γn+e in 𝒰Gal(K). Note that Z1(Sn,Tn/Tn+e) and λ(Z1(P,T)) are both p-groups, and also the subquotient λC(𝒰2Σ) induces a p-group in Aut(C). Recall that 𝒰=ω×θ×𝒰2; by Lemma 4.3, the element θ𝒰 acts trivially in Aut(C), and ω𝒰 acts by multiplication with ω-1. In particular, ω cannot stabilise any non-trivial element in Γn/Γn+e. In summary, if e>0, then 𝒄+Γn+e is non-trivial and Aut(C) is an extension of a p-group by a subgroup of Gal(K). Using the notation of Theorem 4.4, we have proved the following result; recall that Gal(K) is cyclic of order d=p-1 and generated by σ.

Lemma 5.1

If nmax{m,8} and e{1,,n-m}, then

Δn,e*={𝒄Δnthere exists u𝒰 with (u,σ)(𝒄)𝒄modΓn+e}.

It remains to determine which units u𝒰 arise in this setting, and also how to solve the isomorphism problem for the groups defined by elements in Δn,e*. For this purpose, the following two sets are important; they are defined for y; recall that Δn=ΓnΓn+1:

Φn,y={𝒄Γn(ωy,σ)(𝒄)=𝒄},
Λn,y={𝒄Δn(ωy,σ)(𝒄)=𝒄}.

It follows that Λn,y=Φn,yΦn+1,y for every y. Note that the elements of Λn,y are “global” fixed points, while the elements of Δn,e* are fixed points modulo Γn+e. Every global fixed point induces fixed points modulo Γn+e, hence for every e and y we have

Λn,yΔn,e*.

We can now state the main result of this section.

Theorem 5.2

Let 𝐜Δn,e* where nmax{m,8} and e{1,,n-m}.

1. There exist y{(n-2i)moddi=1,,} and 𝒂Λn,y such that Cn,e(𝕊(𝒄))Cn,e(𝕊(𝒂)).

2. If 𝒄Λn,y1 and 𝒅Λn,y2 with y1,y2, then Cn,e(𝕊(𝒄))Cn,e(𝕊(𝒅)) if and only if y1y2modd and 𝒄=(u,1)(𝒅)modΓn+e for some up*.

Our proof of Theorem 5.2 proceeds in several steps and is split up into various lemmas; these are exhibited in the following two subsections.

#### 5.1 Proof of Theorem 5.2 (a)

Throughout this section let nmax{m,8} and e{1,,n-m}.

Lemma 5.3

If 𝐜,𝐝Δn,e*, then

Cn,e(𝕊(𝒄))Cn,e(𝕊(𝒅))

if and only if (u,1)(𝐜)𝐝modΓn+e for some uU.

#### Proof.

By Theorem 4.4, if Cn,e(𝕊(𝒄))Cn,e(𝕊(𝒅)), then

𝒅(w,σb)(𝒄)modΓn+e

for some σbGal(K) and w𝒰. By Lemma 5.1, there exists an element v𝒰 with (v,1)(𝒄)(1,σ)(𝒄)modΓn+e. Write σb=σh for some h{1,,d}, and let u=wvh𝒰. Then

𝒅(w,σh)(𝒄)(w,1)(1,σh)(𝒄)(w,1)(vh,1)(𝒄)(u,1)(𝒄)modΓn+e,

as desired. The converse follows directly from Theorem 4.4. ∎

Lemma 5.4

If 𝐜Δn,e*, then Cn,e(S(𝐜))Cn,e(S(𝐝)) for some 𝐝Δn,e* such that (v,σ)(𝐝)𝐝modΓn+e for some vZp*.

#### Proof.

By Lemma 5.1, there exists u𝒰 with (u,σ)(𝒄)𝒄modΓn+e. We use Lemma 2.2 to decompose u=vx with vkerχ=p* and ximχ. Choose y𝒰 with x-1=χ(y)=yσ(y)-1, and define 𝒅=(y,1)(𝒄). By Theorem 4.4, we have Cn,e(𝒅)Cn,e(𝒄), and the claim follows from

𝒅(y,1)(𝒄)
(y,1)(u,σ)(𝒄)
(y,1)(u,σ)(y-1,1)(𝒅)
(x-1u,σ)(𝒅)
(v,σ)(𝒅)modΓn+e.

In the next lemma we investigate the set Λn,y=Φn,yΔn in more detail. Recall that ω acts by multiplication by ω-1 on Γn, and therefore

(5.1)Φn,y={𝒄Γn(1,σ)(𝒄)=ωy𝒄}.
Lemma 5.5

For each y, the set Φn,y is a Zp-sublattice of Γn; moreover

Γn=Φn,0Φn,d-1.

#### Proof.

The action of 𝒰Gal(K) on Γn defined in Lemma 4.3 extends to an action on the -fold direct product K; in particular, (1,σ) acts via

(x1,,x)(σ(x1),,σ(x)).

By Lemma 2.1, σ:KK is diagonalisable with eigenvalues ω0,,ωd-1, and each eigenspace has dimension 1. Thus, the action of (1,σ) on K is diagonalisable with eigenvalues ω0,,ωd-1 and each eigenspace has dimension . We denote by Vi the eigenspace with eigenvalue ωi in K.

It follows from (5.1) that Φn,y=VyΓn, hence Φn,0Φn,d-1Γn and each Φn,y is a sublattice of Γn. It remains to show the following inclusion: ΓnΦn,0Φn,d-1. For this purpose, consider wΓn and write w=w0++wd-1 with wiVi for i=1,,. Let k be the number of non-zero summands wi in w. We prove by induction on k that each wiΦn,i; then the assertion of the lemma follows.

If k=0 or k=1, then our claim is obviously true. Now suppose that k>1 and choose j with wj0. It follows from Lemma 4.3 that both (1,σ)(w) and (ω-j,1)(w)=ωjw lie in Γn, hence also their difference u=(1,σ)(w)-ωjw lies in Γn. Note that u=u0++ud-1, where each ui=(ωi-ωj)wiVi. By construction, u has at most k-1 non-zero summands ui and, by the induction hypothesis, we have uiΦn,i for each i. Recall that p*=ω(1+pp); this implies that ωi-ωj0modp and hence ωi-ωjp* for all ij. Since ui=(ωi-ωj)wiΦn,i, it follows that wiΦn,i for all ij. In particular, we have w-wj=w0++wj-1+wj+1++wd-1Γn, and so wj=w-(w-wj)Γn as well. Now clearly wjΦn,j, which completes the proof. ∎

The next lemma considers 𝒅Δn,e* and vp* with (v,σ)(𝒅)𝒅modΓn,e, as in Lemma 5.4. Recall that every vp* can be written as

v=ωu(1+px)=ω-y(1+px)

for some u,y{0,,d-1} and xp.

Lemma 5.6

Let 𝐝Δn,e* and vZp* with (v,σ)(𝐝)𝐝modΓn,e, and write v=ω-y(1+px) for some y{0,,d-1} and xZp. There exists 𝐚Λn,y with Cn,e(S(𝐝))=Cn,e(S(𝐚)).

#### Proof.

By definition, 𝒅Δn=ΓnΓn+1; by Lemma 5.5, we can decompose 𝒅=𝒅0++𝒅d-1 with each 𝒅iΦn,i. As (v,σ)(𝒅)𝒅modΓn+e and vp, it follows that

(v,σ)(𝒅)-𝒅=(v-1ω0-1)𝒅0++(v-1ωd-1-1)𝒅d-1Γn+e.

As v-1ωi-1p, Lemma 5.5 yields that (v-1ωi-1)𝒅iΓn+e for each i. Using v=ω-y(1+px), it follows that v-1ωi-1ωi-y-1modp; in particular, v-1ωi-1p* and 𝒅iΓn+e for all iy. We can now choose 𝒂=𝒅yΦn,y; we have shown that 𝒅𝒂modΓn+e, thus

Cn,e(𝕊(𝒅))=Cn,e(𝕊(𝒂)).

As 𝒅Δn, it follows that 𝒂Δn. Hence 𝒂Λn,y as claimed. ∎

Lemma 5.7

We have Λn,y if and only if (1,σ) has an eigenvector in Δ1 with eigenvalue ωy-n+1.

#### Proof.

Let 𝒓Δ1 be an eigenvector of (1,σ) with eigenvalue ωy-n+1. By Lemma 2.1, there exists an eigenvector x𝔭n-1𝔭n of σ with eigenvalue ωn-1. Then x𝒓Δn and

(ωy,σ)(x𝒓)=ω-yσ(x)(1,σ)(𝒓)
=ω-yωn-1xωy-n+1𝒓
=x𝒓,

so x𝒓Λn,y. Conversely, let 𝒄Λn,y. Then 𝒄 is an eigenvector of (1,σ) with eigenvalue ωy. If x𝔭-n+1𝔭-n+2 is an eigenvector of σ with eigenvalue ω-n+1, then x𝒄Δ1 and

(1,σ)(x𝒄)=ω-n+1x(1,σ)(𝒄)
=ω-n+1xωy𝒄
=ωy-n+1x𝒄.

The next lemma is the last result we need for our proof of Theorem 5.2 (a).

Lemma 5.8

The eigenvalues of (1,σ) on Δ1 are {ω-2i+1i=1,,}.

#### Proof.

The main tool in the following proof is Lemma 4.2: with Lemma 2.1 it implies that the eigenvalue of an eigenvector 𝒇Δ1 of (1,σ) is ωval(𝒕), where val(𝒕) is the valuation defined in Section 2. The eigenvalues on Δ1=Γ1Γ2 coincide with the eigenvalues of (1,σ) on the elementary abelian quotient Γ1/Γ2 of rank ; in particular, this number of different eigenvalues is at most . We show that it is exactly . To prove this, we use the notation of Lemma 4.2 and define an isomorphism

ψ:CpΓ1/Γ2,(a1,,a)a1𝒕𝟏++a𝒕+Γ2,

recall that each 𝒕𝒋 has valuation val(𝒕𝒋)=-2j+1. In the following we let (1,σ) act on Cp via ψ; in particular, the action of (1,σ) on Cp is diagonalisable. Our claim is that for each i{1,,} there is, up to scalar multiples, a unique eigenvector (a1,,ai,0,,0)Cp of (1,σ) with ai0, and that this eigenvector has eigenvalue ω-2i+1. We use induction to prove this claim. We note that the assertion of the lemma follows from this claim.

Base case. For the base case consider i=. Since the action of (1,σ) on Cp is diagonalisable, there must be an eigenvector a=(a1,,a)Cp with a0. This eigenvector comes from an eigenvector 𝒇=a1𝒕𝟏++a𝒕+𝒈Δ1 for some 𝒈Δ2. Recall that Δ2=κΔ1(𝔭-2+2). Now

val(a𝒕)=val(a)+val(𝒕)=val(𝒕)=-2+1

and val(𝒈)-2+2 imply that val(𝒇)=-2+1, see Lemma 4.2, whence the eigenvalue of 𝒇 (and a) is ω-2+1. For the uniqueness, consider an eigenvector a=(a1,,a)Cp with a0; as just shown, the eigenvalue is ω-2+1. Without loss of generality, we can assume that a=a. Suppose, for a contradiction, that a and a are linearly independent, so that

b=a-a=(b1,,b-1,0)Cp

is an eigenvector of (1,σ) with eigenvalue ω-2+1. This eigenvector comes from an eigenvector 𝒖=b1𝒕𝟏++b-1𝒕-𝟏+𝒉Δ1 for some 𝒉Δ2. But now val(𝒖)-2+2, so the eigenvalue of b cannot be ω-2+1, a contradiction. (Indeed, if ωval(𝒖)=ω-2+1, then val(𝒖)-2+2 forces val(𝒖)-2+p>0, so 𝒖Δ2 and (b1,,b-1,0)=(0,,0), which is not possible.) This proves that there is, up to scalar multiples, a unique eigenvector of (1,σ) in Cp of the form (a1,,a) with a0, and that the corresponding eigenvalue is ω-2+1.

Induction hypothesis. Our induction hypothesis now is that for each index j{i+1,,} there is, up to scalar multiples, a unique eigenvector of (1,σ) in Cp of the form

vj=(bj,1,,bj,j,0,,0)Cp

with bj,j0, and that the corresponding eigenvalue is ω-2j+1. Note that vj comes from an eigenvector

𝒘𝒋=bj,1𝒕𝟏++bj,j𝒕𝒋+𝒉𝒋Δ1

for some 𝒉𝒋Δ2, with

val(𝒉𝒋)val(bj,1𝒕𝟏++bj,j𝒕𝒋)=val(bj,j𝒕𝒋)=-2j+1.

Existence of eigenvector. It follows from the induction hypothesis that there is an eigenvector

a=(a1,,ai,0,,0)Cp

of (1,σ) with ai0: if not, then there would be a basis of Cp consisting of {vi+1,,v} and i additional eigenvectors each having 0 as k-th entry for all k=i,,; this is not possible as such a set of i vectors cannot be linearly independent. Thus, an eigenvector a as above exists.

Eigenvalue. Our first claim is that the corresponding eigenvalue is ω-2i+1. Note that a comes from an eigenvector

𝒇=a1𝒕𝟏++ai𝒕𝒊+𝒈Δ1

for some 𝒈Δ2; in the following, write r=val(𝒈). If

rval(a1𝒕𝟏++ai𝒕𝒊)=val(ai𝒕𝒊)=val(ai)+val(ai𝒕𝒊)=val(ai𝒕𝒊)=-2i+1,

then val(𝒇)=-2i+1 by Lemma 4.2, and it follows that the eigenvalue is ω-2i+1. It remains to consider the case r<-2i+1, so that the eigenvalue of a is ωr. We show that this is not possible; we achieve this by modifying 𝒈 by our known eigenvectors 𝒘𝒊+𝟏,,𝒘 until we obtain a contradiction. For this purpose, write

𝒈=u1𝒕𝟏++u𝒕Δ2

and let s{1,,} be minimal with val(us𝒕𝒔)=val(𝒈)=r; such an s exists by Lemma 4.2. Note that r=val(us𝒕𝒔)>-2s+1 since us𝔭, hence r+2s-1>0. Since r<-2i+1 by assumption, this implies s{i+1,,}. By the induction hypothesis, we know the existence of the eigenvector

𝒘𝒔=bs,1𝒕𝟏++bs,s𝒕𝒔+𝒉𝒔Δ1

of (1,σ) with eigenvalue ω-2s+1. Let k𝔭r+2s-1𝔭r+2s be an eigenvector of σ with eigenvalue ωr+2s-1. Now 𝒇=𝒇-k𝒘𝒔 is an eigenvector of (1,σ) with eigenvalue ωr, and that both 𝒇 and 𝒇 correspond to

a=(a1,,ai,0,,0)Cp

since k𝔭. In particular,

𝒇=a1𝒕𝟏++ai𝒕𝒊+(𝒈-k𝒘𝒔)
=a1𝒕𝟏++ai𝒕𝒊
+((u1-kbs,1)𝒕𝟏++(us-kbs,s)𝒕𝒔+us+1𝒕𝒔+𝟏++u𝒕)-k𝒉𝒔=:𝒈.

Since both us,kbs,s𝔭r-2s+1𝔭r-2s+2, we can replace k by a suitable scalar multiple of k such that us-kbs,s𝔭r-2s+2, and so val((us-kbs,s)𝒕𝒔)>r; note that val((uj-kbs,j)𝒕𝒋)val(uj𝒕𝒋) for all j. In conclusion, we have found 𝒇=a1𝒕𝟏++ai𝒕𝒊+𝒈 with 𝒈Δ2 such that if 𝒈=u1𝒕𝟏++u𝒕 and s{1,,} is minimal with val(us𝒕𝒔)=val(𝒈)=val(𝒈)=r, then s>s. (Note that 𝒇 has eigenvalue ωr and r<-2i+1, so we must indeed have val(𝒈)=r=val(𝒈).) Now we iterate this argument until we find an eigenvector

𝒇^=a1𝒕𝟏++ai𝒕𝒊+𝒈^

of (1,σ) with eigenvalue ωr and 𝒈^Δ2 such that if 𝒈^=u1^𝒕𝟏++u^𝒕, then val(uj^𝒕𝒋)>r for all j. But then Lemma 4.2 implies that val(𝒈^)>r, a contradiction to ωval(f)=ωr and val(𝒈^)=r. In summary, this proves r-2i+1, hence the eigenvalue of an eigenvector a=(a1,,ai,0,,0) with ai0 must be ω-2i+1.

Uniqueness. Consider a second eigenvector a=(a1,,ai,0,,0)Cp of (1,σ) with ai0; as proved in the previous paragraph, the eigenvalue of a is ω-2i+1. We claim that a and a are linearly dependent. Note that a comes from an eigenvector 𝒇=a1𝒕𝟏++ai𝒕𝒊+𝒈Δ1 for some 𝒈Δ2 with val(𝒈)-2i+1. Similarly, a comes from an eigenvector 𝒇=a1𝒕𝟏++ai𝒕𝒊+𝒈Δ1 for some 𝒈Δ2 with val(𝒈)-2i+1. Suppose, for a contradiction, that a and a are linearly independent. Replacing a by a suitable scalar multiple, we can assume that ai=ai, so that b=a-a=(b1,,bj,0,,0), with bj0 and j<i, is also an eigenvector of (1,σ) with eigenvalue ω-2i+1. This eigenvector comes from 𝒇^=𝒇-𝒇=b1𝒕𝟏++bj𝒕𝒋+𝒈^, where 𝒈^=𝒈-𝒈Δ2 satisfies val(𝒈^)-2i+1; in fact, we must have val(𝒈^)=-2i+1 since otherwise the eigenvalue of b cannot be ω-2i+1. Note that for all u=i+1,, we already found eigenvectors 𝒘𝒖 with eigenvalue ω-2u+1, thus we can use the same construction as in the previous paragraph to obtain from 𝒇^ and 𝒘𝒊+𝟏,,𝒘 an eigenvector 𝒇~=b1𝒕𝟏++bj𝒕𝒋+𝒈~ with eigenvalue ω-2i+1, where 𝒈~Δ2 satisfies val(𝒈~)>-2i+1: this is not possible since 𝒇~ has eigenvalue ω-2i+1, that is, -2i+1=val(𝒇~)=min{-2j+1,val(𝒈~)}, but we have deduced that min{-2j+1,val(𝒈~)}>-2i+1. This contradiction proves that a and a must be linearly dependent. In conclusion, we have proved that, up to scalar multiples, there is a unique eigenvector of (1,σ) of the form (a1,,ai,0,,0) with ai0, and that the corresponding eigenvalue is ω-2i+1. This completes the induction step. ∎

Lemmas 5.7 and 5.8 yield the following corollary.

Corollary 5.9

We have Λn,y if and only if yn-2imodd for some i=1,,.

We can now prove Theorem 5.2 (a).

#### Proof of Theorem 5.2 (a).

Let 𝒄Δn,e*. By Lemmas 5.4 and 5.6, there exist y{0,,d-1} and 𝒂Λn,y such that

Cn,e(𝕊(𝒄))Cn,e(𝕊(𝒂)).

Since Λn,y, it follows from Corollary 5.9 that yn-2imodd for some i{1,,}, as claimed. ∎

#### 5.2 Proof of Theorem 5.2 (b)

Again, we assume that nmax{m,8} and e{1,,n-m}. Recall the map χ:𝒰𝒰, uuσ(u)-1, which is discussed in Lemma 2.2.

Lemma 5.10

Let 𝐜Λn,y1 and 𝐝Λn,y2 with y1,y2Z. If Cn,e(S(𝐜))Cn,e(S(𝐝)), then y1=y2; if uU with 𝐝(u,1)(𝐜)modΓn+e as in Lemma 5.3, then 𝐜 and 𝐝 are fixed points of (χ(u),1) modulo Γn+e.

#### Proof.

It is shown in Lemma 5.3 that 𝒅(u,1)(𝒄)modΓn+e for some u𝒰. This implies that

𝒅(ωy2,σ)(𝒅)
(ωy2,σ)(u,1)(𝒄)
(ωy2,σ)(u,1)(ωy1,σ)-1(𝒄)
(ωy2,σ)(u,1)(ωy1,σ)-1(u,1)-1(𝒅)
(ωy2-y1σ(u)u-1,1)(𝒅)
(ωy2-y1χ(u)-1,1)(𝒅)modΓn+e.

By Theorem 4.4, the element (ωy2-y1χ(u)-1,1) yields an element of the group Aut(Cn,e(𝒅)), and the discussion of the automorphism groups of skeleton groups (in the beginning of Section 5) forces that y2y1modd. Since we have y1,y2{0,,d-1}, this yields y1=y2. In turn, this implies that 𝒅 (and, by duality also 𝒄) are fixed points of (χ(u)-1,1) modulo Γn+e. ∎

Lemma 5.11

Let 𝐜,𝐝Λn,y with yZ. If Cn,e(S(𝐜))Cn,e(S(𝐝)), then 𝐜(s,1)(𝐝)modΓn+e for some sZp*.

#### Proof.

Recall that 𝒄,𝒅Λn,yΔn,e*. Thus, by Lemma 5.3, there exists u𝒰 with (u,1)(𝒄)𝒅modΓn+e. By Lemma 5.10, both 𝒄 and 𝒅 are fixed points under (χ(u),1). It follows from Lemma 2.2 that the restriction χ|imχ induces an automorphism of imχ; write u=sw with skerχ=p* and wimχ. We want to show that (w,1)(𝒅)𝒅modΓn+e since this implies that

𝒄(u,1)(𝒅)(s,1)(w,1)(𝒅)(s,1)(𝒅)modΓn+e,

which proves the lemma.

We start with a more general observation: Let v𝒰 and suppose (v,1) maps the fixed point 𝒅 of (ωy,σ) to another fixed point (v,1)(𝒅) of (ωy,σ) modulo Γn+e. Then it follows from

(v,1)(𝒅)(ωy,σ)(v,1)(𝒅)(σ(v),1)(ωy,σ)(𝒅)(σ(v),1)(𝒅)modΓn+e

that such a (v,1) has an image (χ(v),1) which acts trivially on 𝒅 modulo Γn,e, and by duality also on 𝒄 modulo Γn,e.

Since (u,1) maps the fixed point 𝒅 to the fixed point 𝒄, it follows that (χ(u),1) stabilises 𝒅 modulo Γn+e. Note that χ(u)=χ(w), hence also (χ(w),1) stabilises 𝒅 modulo Γn+e. Now we iterate this argument: since (χ(w),1) maps the fixed point 𝒅 of (ωy,1) to the fixed point 𝒅 of (ωy,1), modulo Γn+e, it follows from the general observation that (χ2(w),1) stabilises 𝒅 modulo Γn+e. By induction, (χi(w),1) stabilises 𝒅 modulo Γn+e for every i1.

There is j2 such that 𝒰j=1+𝔭j acts trivially on Γn/Γn+e: for example, choose j large enough such that 𝔭jpx𝒪 for some x with pxΓnΓn+e. It follows from the definition that χ stabilises 𝒰j. Since 𝒰j𝒰2, the proof of Lemma 2.2 shows that 𝒰j=ker(χ|𝒰j)×im(χ|𝒰j); this implies that χ induces an automorphism of J=(imχ)/(imχ|𝒰j), which we denote by ψ. Since J is a finite group, it follows that ψ has finite order, say t.

Recall that w as above lies in imχ. If w lies in 𝒰j, then (w,1) acts trivially on 𝒅 modulo Γn+e, and there is nothing to show. If w𝒰j, then its coset c=wim(χ|𝒰j) in J is non-trivial, and ψt(c)=c follows. But this means that χt(w)=wr for some r𝒰j. As shown above, χt(w)=wr stabilises 𝒅 modulo Γn+e. Since r𝒰j acts trivially on Γn/Γn+e, it follows that w stabilises 𝒅 modulo Γn+e. ∎

We can now prove Theorem 5.2 (b).

#### Proof of Theorem 5.2(b).

Let 𝒄Λn,y1 and 𝒅Λn,y2. If there exists an element up* with 𝒄(u,1)(𝒅)modΓn+e, then we have Cn,e(𝕊(𝒄))Cn,e(𝕊(𝒅)) by Lemma 5.3. For the converse, suppose that Cn,e(𝕊(𝒄))Cn,e(𝕊(𝒅)). Then y1y2modd follows from Lemma 5.10, and 𝒄=(u,1)(𝒅)modΓn+e for some up* follows from Lemma 5.11. ∎

### 6 The ramification levels in the skeleton 𝒮n∗

We apply Theorem 5.2 to prove the explicit description of the skeleton 𝒮n as suggested in Theorem 1.1. First we consider the groups of depth 1.

Theorem 6.1

Let nmax{8,m}. There are groups Gn,1,,Gn, at depth 1 in Sn; these are obtained as Gn,i=Cn,1(S(𝐜i)), where 𝐜i is a fixed point of (ωji,σ) with ji=n-2imodd.

### Proof.

It follows from Theorem 5.2 (a) that the groups at depth 1 in 𝒮n can be constructed as Cn,1(𝒄) with 𝒄Λn,y for y=0,,d-1. It is shown in Corollary 5.9 that Λn,y if and only if y=yi=n-2imodd for some index i{1,,}; this allows us to construct groups Gn,i=Cn,1(𝒄𝒊), where 𝒄𝒊Λn,yi for i=1,,. Theorem 5.2 (b) shows that Gn,iGn,j if and only if i=j. It follows from Lemmas 5.7 and 5.8 that the eigenspace of each (ωyi,σ) in Γn/Γn+1 has dimension 1. Thus the elements in Λn,yi admit exactly one isomorphism type of skeleton group for each i=1,,. This completes the proof. ∎

Now we consider the remaining part of the skeleton.

Theorem 6.2

Let i{1,,} and let H be a descendant of Gn,i at depth e{1,,n-m-1} in Sn, where Gn,i is as in Theorem 6.1. The group H has p immediate descendants in Sn if and only if emodd{2,4,,d-2}{d-2i}; otherwise, H has one immediate descendant in Sn.

### Proof.

We investigate the descendants of Gn,i for a fixed i{1,,}. By Theorems 5.2 and 6.1, our aim is to determine the orbits and stabilisers of p* acting on

Πn,j,e={𝒄+Γn+e𝒄Λn,j},

where j=n-2imodd. In the following, let 𝒞={0,2,,d-2}{d-2i}.

We first consider stabilisers; recall that p=ω(1+pp), and note that up* acts on 𝒅Λn,j via

(u,1)(𝒅)=u-1𝒅modΓn+e.

Thus (u,1)(𝒅)𝒅modΓn+e if and only if u1+pe¯ with e¯=e/d; this implies that Stabp*(𝒅+Γn+e)=1+pe/dp. Note that the stabiliser depends on e, but not on 𝒅. In particular, it follows that each p*-orbit has the same size, namely dpe¯-1.

Next we show that

|Πn,j,e|={|Πn,j,e+1|if emodd𝒞,p|Πn,j,e+1|otherwise.

To prove this, note that |Πn,j,e|<|Πn,j,e+1| if and only if there exist 𝒅,𝒆Λn,j with 𝒅𝒆modΓn+e and 𝒅𝒆modΓn+e+1. This holds if and only if there exists a fixed point 𝒓Γn+eΓn+e+1 of (ωj,σ) with 𝒅=𝒄+𝒓. By Corollary 5.9, such a fixed point exists if and only if jn+e-2imodd for some i{1,,}. Since jn-2imodd, such an i exists if and only if e2(i-i)modd. A straightforward computation shows that

𝒞={2(i-i)moddi=1,,}.

Thus, in summary, |Πn,j,e|>|Πn,j,e+1| if and only if emodd𝒞; in this case, |Πn,j,e|=p|Πn,j,e+1| follows since the eigenspace of (ωj,σ) on Γn+e/Γn+e+1 has dimension 1.

It remains to consider e0modd. In this case, the size of the action domain grows by p, but also the size of the orbits grows by p, that is, the number of orbits remains stable; in other words, H has a single immediate descendant in 𝒮n. If emodd𝒞{0}, then the size of the action domain grows by p, but the size of the orbit stays the same; in other words, H has p immediate descendants in 𝒮n. ∎

Communicated by Evgenii I. Khukhro

Funding statement: This research was supported by a Go8-DAAD Joint Research Co-operation Scheme, project “Groups of Prime-Power Order and Coclass Theory”. The second author was also supported by an ARC DECRA (Australia), project DE140100088.

### A Historical notes

A serious problem for classifying finite p-groups is that the number of isomorphism types of p-groups of order pn grows exponentially with n; for large n, this makes a classification by order an impossible task. A more promising approach to bring structure into the realm of p-groups is to consider finite p-groups by coclass, where the coclass of a finite group of order pn and nilpotency class c is defined as n-c. Note that the p-groups of maximal class are exactly the p-groups of coclass 1. Initiated by Leedham-Green and Newman [12] in 1980, this program is still an active area of research (cf. the recent work [18, 5, 4, 3, 9, 2, 7]), which has led to deep and interesting results; we refer to the book of Leedham-Green and McKay [11] for more details and references.

A main tool in coclass theory is the coclass graph 𝒢(p,r) associated with the p-groups of coclass r. As for maximal class, the vertices of 𝒢(p,r) are identified with isomorphism type representatives of the considered groups, and there is an edge GH if and only if G is isomorphic to H/γ(H), where γ(H) is the last non-trivial term of the lower central series of H. It is known that 𝒢(p,r) can be partitioned into a finite set of isolated groups, and a finite collection of coclass trees: a coclass tree is an infinite tree 𝒯 which has a unique infinite path G1G2 starting at its root. The branchn of 𝒯 is the subtree of 𝒯 generated by all descendants of Gn which are not descendants of Gn+1; thus, every coclass tree can be partitioned into its branches, which are connected via the infinite path.

The main focus in coclass theory currently is to understand the structure of 𝒢(p,r). The aim of this appendix is to provide more details on known periodicity results for 𝒢(p,r), thereby putting our main results into context. We do not claim to present a complete historical account on existing results.

#### A.1 Coclass theory

The origins of coclass theory lie in the study of p-groups of maximal class. This study was initiated by Wiman [19] in 1952, and the first major results are due to Blackburn [1] in 1958. In particular, Blackburn obtained a complete classification of the 2- and 3-groups of maximal class. Motivated by Blackburn’s success, the p-groups of maximal class became a well-studied type of p-groups and, as a generalisation, Leedham-Green and Newman defined the coclass of a p-group in their 1980 paper [12]. Now coclass theory started out in two directions.

First, Leedham-Green and Newman related p-groups of a fixed coclass to certain extensions of uniserial space groups, so called pro-p-groups of fixed coclass. Their investigations culminated in the formulation of five Coclass Conjectures, called Conjecture A–E, where Conjecture A is the strongest since it implies Conjectures B–E. Many authors contributed to a proof of these conjectures, and the final proof of Conjecture A was found independently by Leedham-Green and Shalev, both in 1994. For details and references we refer to [11] and also the book of Dixon, du Sautoy, Mann and Segal [6, p. 265]. We remark that it is Conjecture D which implies that each coclass graph 𝒢(p,r) has finitely many coclass trees.

Second, between 1976 and 1984, Leedham-Green and McKay published a series of papers on p-groups of maximal class, see [10] and the references given there. The concept of skeleton groups (groups in the coclass graph which are defined by certain homomorphisms, cf. Section 4 for coclass 1) has its roots in these papers. It was proved later that these skeleton groups essentially determine the general structure of a coclass graph, cf. [11, Section 11], which underpins the importance of the skeletons.

Motivated by further promising computer experiments, the focus of coclass theory then turned to the investigation of the detailed structure of coclass graphs. The next section describes the main highlights of the last two decades.

#### A.2 Periodicities in coclass graphs

In general, 𝒯 denotes a coclass tree with branches n in some specified coclass graph 𝒢(p,r). For an integer k>0 let n(k) be the pruned subtree of n generated by all groups at depth at most k in n.

For p{2,3}, Blackburn proved that the branches of the coclass tree in 𝒢(p,1) satisfy nn+p-1 for all large enough n, that is, 𝒢(p,1) is virtually periodic.

Newman [14], and later Dietrich, Eick and Feichtenschlager [5], did extensive computer experiments for 𝒢(5,1), which suggest that n(n-1)n+4(n-1) and n+4n+4(n-1)nn(n-5) for all large enough n. Eventually Dietrich [3] proved that indeed

n(n-4)n+4(n-4)andn+4(n)n+4(n-4)n(n-4)n(n-8)

for all large enough n; the proof that n+4Bn+4(n)nn(n-4) for all large enough n is currently still missing. Nevertheless, these periodicity results describe 𝒢(5,1) almost completely.

The investigations by Newman and by Leedham-Green and McKay already showed that the p-groups of maximal class are significantly more difficult to classify for p7. Their analysis revealed that the structure of 𝒢(p,1) is very complicated, and that a complete classification seems a highly non-trivial task. As a special case, Leedham-Green and McKay studied a subtree of the coclass tree in 𝒢(p,1) consisting of certain capable “1-parameter groups”. Using the language of Section 4, these are the skeleton groups of the type Cn,e(𝕊(𝒄)), where 𝒄=(c1,,c) with exactly one non-zero ci. These subtrees of 1-parameter groups have finite widths if p5mod6, cf. the comment on [10, p. 299].

Newman and O’Brien [15] investigated the graph 𝒢(2,r) for arbitrary r; their extensive computations led to the conjecture that each coclass tree in 𝒢(2,r) is virtually periodic, that is, there is an integer d1 such that nn+d for all large enough n.

The first periodicity theorem for general coclass graphs 𝒢(p,r) was established independently by du Sautoy [18] and Eick and Leedham-Green [8]: they proved that for every coclass tree with branches 1,2, and every integer k>0, there exists d1 such that n(k)n+d(k) for all large enough n. The results by Eick and Leedham-Green [8] yield further that the virtual periodicity of a coclass tree translates to a classification of the groups in this tree in terms of finitely many parametrised group presentations. It is known that this periodicity pattern is capable of describing the complete graph 𝒢(p,r) if and only if p=2 or (p,r){(2,1),(3,1)}. In all other cases, there exist coclass trees which have branches of arbitrarily large depth and a second periodic pattern is required to describe the growth of these branches.

Dietrich [4, 3] considered 𝒢(p,1) in detail for p5mod6; this work is the first analysis of coclass trees of infinite width. In particular, the results in [3] led to the aforementioned (almost complete) classification of 𝒢(5,1), which has finite width. For p>5 the coclass tree in 𝒢(p,1) has infinite width, and the main result can be described as follows: First, there is an isomorphism of pruned branches n(n-2p+8)n+p-1(n-2p+8) for all large enough n. Second, if G is a capable group at depth n-2p+8 in n+p-1 and if the automorphism group of its (p-1)-step parent H is a p-group, then 𝒟p-1(G)𝒟p-1(H), where 𝒟p-1(K) is the subtree generated by all descendants of K of distance at most p-1 to K. This second periodicity result describes the growth of the branches in some cases; it is a local result since it requires knowledge of the structure of the group and its (p-1)-step parent. It is known, however, that “almost all” p-groups have a p-group as automorphism group, hence the results in [3] can be used to describe large parts of 𝒢(p,1). We conclude this paragraph with two comments: First, the results in [3] are slightly more general than described here: a second periodicity result can also be formulated for groups whose (p-1)-step parent does not have a p-group as automorphism group; instead, the requirement is that the group in question has bounded distance to a maximal path in n whose groups have automorphism group orders with constant p-part; we refer to [3, Theorem 1.3] for more details. Note that the groups we consider in Theorem 1.1 satisfy this condition, which relates our work to the approach in [3]. Second, the maybe surprising restriction to p5mod6 comes from underlying problems in p-adic number theory, already discussed in [10].

Eick, Leedham-Green, Newman and O’Brien [9] have investigated 𝒢(3,2) in detail. More precisely, they have studied the skeletons of each of the sixteen coclass trees of 𝒢(3,2). Each of these skeleton groups is a 1-parameter group (of coclass 2), and the trees have finite width. Based on their computations and the aforementioned existing results, Conjecture W in [9] suggests a construction of 𝒢(p,r) from a finite subgroup. We briefly sketch this conjecture here; for full details see [9, Section 9]. Let 𝒯 be a coclass tree in 𝒢(p,r) with branches 1,2, of arbitrarily large depths. Choose k1; the results in [8] imply that there exist d1 and l such that Bl(k)Bl+id(k) for all i. For a group PBl+id(k) denote by P¯ the group in l(k) under a suitable graph isomorphism. Conjecture W now states that if one chooses k and l large enough, then there is a map ν from the groups of depth k in l to the groups of depth k-d in l such that the following holds: If P has depth k in l+id, then 𝒟(P)𝒟(Q), where Q is the group at depth k-d in l+(i-1)d corresponding to Q¯=ν(P¯), and 𝒟(K) is the subtree generated by all descendants of K. This conjecture is illustrated in Figure 3.

### Figure 3

An illustration of Conjecture W.

In conclusion, with the exception of the local periodicity results in [3], all known periodicity results are for pruned subgraphs consisting of skeleton groups in coclass trees of finite width. Our Theorem 1.1 is the first result for such trees of infinite widths, and the first significant evidence supporting Conjecture W in this case.

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