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Publicly Available Published by De Gruyter September 14, 2016

Finite p-groups of maximal class with โ€˜largeโ€™ 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 ๐’ข associated with the p-groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph ๐’ขโˆ— of ๐’ข associated with those p-groups of maximal class whose automorphism group orders are divisible by p-1. We describe the broad structure of ๐’ขโˆ— 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 ๐’ขโˆ— for arbitrary pโ‰ฅ7. 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 graph๐’ข associated with the p-groups of maximal class: the vertices of ๐’ข are identified with isomorphism type representatives of the considered groups, and there is an edge Gโ†’H if and only if H/ฮณโข(H)โ‰…G, where ฮณโข(H) is the last non-trivial term of the lower central series of H. Investigating the structure of ๐’ข is an approach towards a detailed understanding (and thus towards a possible classification) of the associated groups.

It is well known that ๐’ข 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 S2โ†’S3โ†’โ‹ฏ 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 ๐’ข in more detail. If there is a path of length k from a group G to a group H in ๐’ข, 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 nโˆˆโ„• the branchโ„ฌn of ๐’ข is the subtree generated by all descendants of Sn which are not descendants of Sn+1. Note that every branch โ„ฌn is a finite tree with root Sn, and the whole coclass tree of ๐’ข is partitioned into its branches, which are connected via the infinite path, see Figure 1. In conclusion, the structure of ๐’ข is determined by the structure of its branches.

Figure 1 The coclass graph ๐’ข${\mathcal{G}}$ with its coclass tree and branches โ„ฌn${\mathcal{B}_{n}}$.

Figure 1

The coclass graph ๐’ข with its coclass tree and branches โ„ฌn.

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 kโˆˆโ„• the pruned branch โ„ฌnโข(k) is the subtree of โ„ฌn generated by all groups of depth at most k in โ„ฌn. It was proved independently by du Sautoy [18] and Eick and Leedham-Green [8] that for each kโˆˆโ„• there exists lโข(k)>0 such that โ„ฌn+p-1โข(k)โ‰…โ„ฌnโข(k) for all nโ‰ฅlโข(k). Thus the pruned branches eventually repeat periodically; we call this the first periodicity.

If pโˆˆ{2,3}, then all branches in ๐’ข 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=2โขp-8, see for example [4, Theorem 1.2]. Hence, the first periodicity is not capable of describing ๐’ข 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 ๐’ขcapable if it is not a leaf, and we define the skeleton๐’ฎn as the subtree of โ„ฌn generated by all capable groups at depth at most n-m in โ„ฌn. 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 ๐’ข has finite width if the number of groups of fixed depth in โ„ฌn is bounded by a constant independent of n. It is known that ๐’ข has finite width if and only if pโ‰ค5. 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 pโ‰ฅ7. Here the graph ๐’ข 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 ๐’ข. 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 nโ‰ฅk the branch โ„ฌn+p-1 can be constructed from โ„ฌn 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 pโ‰ฅ7, which is fixed throughout this paper, and define constants

d=p-1,โ„“=p-32,m=2โขp-8.

We use the notation ๐’ข, โ„ฌn, ๐’ฎn defined above. The aim of this paper is to study the subgraph ๐’ขโˆ— of ๐’ข generated by all groups whose automorphism group order is divisible by d. More precisely, we investigate ๐’ขโˆ— 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 โ„ฌnโˆ— and ๐’ฎnโˆ— the subgraphs of โ„ฌn and ๐’ฎn contained in ๐’ขโˆ—; these are both subtrees of โ„ฌn. Our first result is a complete determination of the skeletons ๐’ฎnโˆ—; see Section 6 for its proof.

Theorem 1.1

Let nโ‰ฅmaxโก{m,8}.

  1. The skeleton ๐’ฎnโˆ— has โ„“ groups at depth 1 ; we denote these by Gn,1,โ€ฆ,Gn,โ„“.

  2. Let H be a descendant of Gn,i at depth e<n-m in ๐’ฎnโˆ—. Then H has p immediate descendants in ๐’ฎnโˆ— if and only if (emodd)โˆˆ{2,4,โ€ฆ,d-2}โˆ–{d-2โขi}; otherwise, H has one immediate descendant in ๐’ฎnโˆ—.

Theorem 1.1 exhibits that each skeleton ๐’ฎnโˆ— consists of โ„“ 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 nโ‰ฅmaxโก{m,8} and e<n-m. The number of groups at depth e in Snโˆ— is at least โ„“โขpโŒŠe/dโŒ‹โข(โ„“-1); in particular, Gโˆ— has infinite width.

We have determined โ„ฌnโˆ— for p=7 and 10โ‰คnโ‰ค18 by computer, cf. Section 1.2. Based on this, we formulate a conjectural description for โ„ฌnโˆ— for arbitrary pโ‰ฅ7. To describe this conjecture, we define the twig๐’ฒGโˆ— of a group G in ๐’ฎnโˆ—: this is the subtree of โ„ฌnโˆ— with root G containing all descendants of G which are not descendants of any proper descendant of G in ๐’ฎnโˆ—. Thus each group of โ„ฌnโˆ— is contained in exactly one twig and the twigs of โ„ฌnโˆ— are connected by the skeleton of โ„ฌnโˆ—. We continue to denote the groups of depth 1 in a skeleton ๐’ฎnโˆ— by Gn,1,โ€ฆ,Gn,โ„“ and we assume that these are sorted via Theorem 1.1โ€‰(b). Note that every group G of depth at least 1 in ๐’ฎnโˆ— has exactly one of the groups Gn,i as parent.

Conjecture 1.3

There exists lโ‰ฅm such that for all nโ‰ฅl and for each G of depth e at least 1 in ๐’ฎnโˆ— with parent Gn,i the following holds.

  1. If G is not a leaf in ๐’ฎnโˆ—, then the isomorphism type of the graph ๐’ฒGโˆ— depends on the index i, on emodd, and on nmodd only.

  2. If G is a leaf in ๐’ฎnโˆ—, then there exists a group Gยฏ with parent Gn-d,i at depth e-d in ๐’ฎn-dโˆ— with ๐’ฒGโˆ—โ‰…๐’ฒGยฏโˆ—.

Conjecture 1.3 suggests that there are three types of twigs: the twigs of the leaves in ๐’ฎnโˆ—, the twigs of the roots of ๐’ฎnโˆ—, and the twigs of the other skeleton groups. Note that, by definition, ๐’ฒGโˆ— has depth at most 1 for every group G in ๐’ฎnโˆ— which is not a leaf. We choose l large enough such that the first periodicity holds for all groups of depth 1 in โ„ฌnโˆ—; then the twigs of the roots in ๐’ฎnโˆ— behave periodically for all nโ‰ฅl and thus there are at most l+d different twigs of roots in ๐’ขโˆ—. Conjecture 1.3 suggests that for nโ‰ฅl there are only โ„“โขd2 different twigs in ๐’ฎnโˆ— for groups that are neither roots nor leaves. The twigs of the leaves of ๐’ฎnโˆ— are trees of depth at most 2โขm+3. Conjecture 1.3 suggests that there are finitely many different twigs of skeleton-leaves; more precisely, if the skeleton ๐’ฎnโˆ— has wn leaves, then Conjecture 1.3 proposes that there are at most wl+1+โ€ฆ+wl+d different twigs arising for the leaves in all ๐’ฎnโˆ— with nโ‰ฅl.

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 โ„ฌ10โˆ—,โ€ฆ,โ„ฌ16โˆ— 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 ๐’ฎnโˆ—.

Figure 2 The branches โ„ฌnโˆ—${{\mathcal{B}}^{\ast}_{n}}$ (and skeleton ๐’ฎnโˆ—${{\mathcal{S}}^{\ast}_{n}}$)
for p=7${p=7}$ and n=10,โ€ฆ,16${n=10,\dots,16}$.

Figure 2

The branches โ„ฌnโˆ— (and skeleton ๐’ฎnโˆ—) for p=7 and n=10,โ€ฆ,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 ๐’ฎn, and we consider their isomorphism problem and automorphism groups. In Section 5 we then investigate the skeleton groups in ๐’ฎnโˆ— 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, โ„šp and โ„คp denote the field of p-adic rational numbers and ring of p-adic integers, respectively. The p-th cyclotomic polynomial

1+x+โ€ฆ+xp-1โˆˆโ„špโข[x]

is irreducible; we consider a fixed root ฮธ and define K=โ„špโข(ฮธ), so that K is a field extension of degree d=p-1 over โ„šp, with โ„šp-basis {1,ฮธ,โ€ฆ,ฮธd-1}. For aโˆˆโ„ค with pโˆคa the field automorphism ฯƒa:Kโ†’K is defined by ฯƒaโข(ฮธ)=ฮธa. The Galois group of K is Galโข(K)={ฯƒaโˆฃ1โ‰คaโ‰คd}; it is cyclic and we fix a generator ฯƒ=ฯƒk. The equation order ๐’ช=โ„คpโข[ฮธ] is the maximal order of K; it has {1,ฮธ,โ€ฆ,ฮธd-1} as โ„คp-basis, and a unique maximal ideal ๐”ญ=(ฮธ-1). We abbreviate ฮบ=ฮธ-1, so that (ฮบm)=๐”ญm for mโˆˆโ„ค; this defines a series of ideals through ๐’ช. For zโˆˆโ„ค and nโˆˆโ„• we denote by (๐”ญz)n the direct sum of n copies of ๐”ญz (and not the ideal ๐”ญnโขz). For each non-zero wโˆˆK there exist unique zโˆˆโ„ค and a unit uโˆˆโ„คpโข[ฮธ]โˆ— such that w=ฮบzโขu; we call valโข(w)=z the valuation of w. Note that if v,wโˆˆK are non-zero, then valโข(vโขw)=valโข(v)+valโข(w). We extend this definition to non-zero n-tuples ๐’—โˆˆKn, so that valโข(๐’—)=z if and only if ๐’—โˆˆ(๐”ญz)nโˆ–(๐”ญz+1)n.

2.1 Eigenvalues

The generator ฯƒ of the Galois group of K can be considered as a โ„šp-linear map of K. The next lemma determines its eigenvalues; it is proved in [10, Lemma 2.3] for zโ‰ฅ0, but the same proof holds for all zโˆˆโ„ค.

Lemma 2.1

The eigenvalues of the Qp-linear map ฯƒ:Kโ†’K are ฯ‰0,โ€ฆ,ฯ‰d-1 and each eigenspace has dimension 1. If wโˆˆK with valโข(w)=z is an eigenvector of ฯƒ, then the corresponding eigenvalue is ฯ‰z. For every zโˆˆZ there exists an eigenvector w of ฯƒ with valโข(w)=z.

2.2 The group of units

Let ๐’ฐ be the unit group of ๐’ช. For iโ‰ฅ2 define ๐’ฐi=1+๐”ญi, and let ฯ‰ be a primitive d-th root of unity in โ„คp; we assume throughout that ฯ‰ is chosen such that ฯ‰โ‰กkmodp where k is defined by the fixed generator ฯƒ=ฯƒk of Galโข(K). It is shown in [13, Satz II.5.3] that

๐’ฐ=ใ€ˆฯ‰ใ€‰ร—ใ€ˆฮธใ€‰ร—๐’ฐ2.

The unit group of โ„คp is โ„špโˆฉ๐’ฐ, that is, โ„คpโˆ—=ใ€ˆฯ‰ใ€‰ร—(1+pโขโ„คp). 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 ฯ‡:Uโ†’U, uโ†ฆuโขฯƒโข(u)-1, is a group homomorphism with kerโกฯ‡=Zp* and U=kerโกฯ‡ร—imโกฯ‡.

Proof.

Since ฯƒ generates Galโข(K), the fixed points of ฯƒ in K are exactly the elements of the subfield โ„šp of K, hence kerโกฯ‡=๐’ฐโˆฉโ„šp=โ„คpโˆ—. Next, we consider the restriction of ฯ‡ to ๐’ฐ2=1+๐”ญ2. As shown in [13, Satz II.5.5 and p.โ€‰146], the additive group of ๐”ญ2 is isomorphic to the multiplicative group ๐’ฐ2 via

exp:(๐”ญ2,+)โ†’(๐’ฐ2,โ‹…),xโ†ฆโˆ‘k=0โˆžxkk!.

As this exponential map is compatible with the action of ฯƒ, we can translate ฯ‡ to a map ฯˆ:๐”ญ2โ†’๐”ญ2, xโ†ฆx-ฯƒโข(x). Since ฯƒ is a diagonalisable โ„คp-linear map on ๐”ญ2 with eigenvalues ฯ‰0,ฯ‰1,โ€ฆ,ฯ‰d-1, the map ฯˆ is diagonalisable with eigenvalues 0,1-ฯ‰1,โ€ฆ,1-ฯ‰d-1. Now note that if iโˆˆ{1,โ€ฆ,d-1}, then 1-ฯ‰iโ‰ข0modp, so 1-ฯ‰iโˆˆโ„คpโˆ—; moreover, Lemma 2.1 implies that there exists an eigenvector of ฯˆ in ๐”ญ2โˆ–๐”ญ3. In conclusion, we have shown that

๐”ญ2=kerโกฯˆโŠ•imโกฯˆ,

and hence ๐’ฐ2 is the direct product of kerโกฯ‡ and imโกฯ‡ restricted to ๐’ฐ2. Finally, note that ฯ‡โข(ฯ‰)=1 and ฯ‡โข(ฮธ)=ฮธ1-k. Thus the result follows from the decomposition of ๐’ฐ as ๐’ฐ=ใ€ˆฯ‰ใ€‰ร—ใ€ˆฮธใ€‰ร—๐’ฐ2. โˆŽ

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=(๐’ช,+) the additive group of the ring ๐’ช, and let P be the cyclic group of order p generated by ฮธ. We let P act on T by multiplication.

Lemma 3.1

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

The unique maximal S-invariant series through T is T=T1>T2>โ‹ฏ, where each Ti=(๐”ญi-1,+). Moreover, if iโ‰ฅ2, then Ti=ฮณiโข(S) is the i-th term in the lower central series of S. For iโˆˆโ„• define Si=S/ฮณiโข(S), so that S2โ†’S3โ†’โ‹ฏ is the unique maximal infinite path in ๐’ข. 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 S2โ†’S3โ†’โ‹ฏ is contained in ๐’ขโˆ—.

Lemma 3.2

Writing elements of S as tuples (t,ฮธi) with tโˆˆT and iโˆˆZ, the following hold:

  1. The natural restriction ฯ€:Autโข(S)โ†’Autโข(T), ฮฑโ†ฆฮฑ|T, satisfies

    imโกฯ€โ‰…๐’ฐโ‹ŠGalโข(K)โ€ƒ๐‘Ž๐‘›๐‘‘โ€ƒkerโกฯ€โ‰…Z1โข(P,T).

    A preimage of (u,ฯƒb)โˆˆ๐’ฐโ‹ŠGalโข(K) under ฯ€ is given by

    ฮฑโข(u,b):Sโ†’S,(t,ฮธi)โ†ฆ(uโขฯƒbโข(t),ฮธiโขb).

    The kernel of ฯ€ is generated by {ฮฑโข(1),ฮฑโข(ฮธ),โ€ฆ,ฮฑโข(ฮธd-1)}, where for sโˆˆT we define

    ฮฑโข(s):Sโ†’S,(t,ฮธi)โ†ฆ(t+(1+ฮธ+โ€ฆ+ฮธi-1)โขs,ฮธi).
  2. If iโ‰ฅ4, then the natural projection Autโข(S)โ†’Autโข(Si) is surjective and |Outโข(Si)|=pi-2โขd2.

4 The skeleton groups in ๐’ฎn

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 TโˆงT: this is the โ„คpโขP-module generated by sโˆงt with s,tโˆˆT such that for all s,sโ€ฒ,tโˆˆT and zโˆˆโ„คp the following holds: sโˆงs=0, hence sโˆงt=-(tโˆงs), and zโข(sโˆงt)=(zโขs)โˆงt=sโˆง(zโขt), and (sโˆงt)+(sโ€ฒโˆงt)=(s+sโ€ฒ)โˆงt. The group P acts diagonally on TโˆงT, which defines the โ„คpโขP action on TโˆงT.

In the following let nโ‰ฅmaxโก{m,8} and eโˆˆ{0,โ€ฆ,n-m}. Every surjective homomorphism fโˆˆHomPโข(TโˆงT,Tn) defines an associative multiplication on T/Tn+e via

(t+Tn+e)โˆ˜(s+Tn+e)=t+s+12โขfโข(tโˆงs)+Tn+e.

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

Cn,eโข(f)=Mn,eโข(f)โ‹ŠP;

these groups are called constructible in [11]. Each group Cn,eโข(f) is an extension of the natural Sn-module Tn/Tn+e by the group Sn on the infinite path of ๐’ข; in particular, it is a group of depth e in the skeleton ๐’ฎn. By [4, Theorem 1.3], the groups Cn,eโข(f) are exactly the groups in the skeleton ๐’ฎn.

4.1 The structure of HomPโข(TโˆงT,T)

The structure of HomPโข(TโˆงT,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, TโˆงT=FโŠ•Z, where F is a โ„คpโขP-module of rank โ„“=(p-3)/2 generated by ฮบiโˆงฮบi-1 with i=1,โ€ฆ,โ„“, and Z is a free โ„คp-module of rank 1. Let ฮดiโขj denote the Kronecker-delta and recall that ฯƒa is an element of Galโข(K) for pโˆคa. For iโˆˆ{1,โ€ฆ,โ„“} we define ๐•‹iโˆˆHomPโข(TโˆงT,T) via

๐•‹iโข(ฮบjโˆงฮบj-1)=ฮดiโขjโ€ƒandโ€ƒ๐•‹iโข(z)=0โข for โขzโˆˆZ.

For 2โ‰คaโ‰คโ„“+1 we define ๐•ŠaโˆˆHomPโข(TโˆงT,T) via

๐•Šaโข(xโˆงy)=ฯƒaโข(x)โขฯƒ1-aโข(y)-ฯƒ1-aโข(x)โขฯƒaโข(y).

For an โ„“-tuple ๐’„=(c1,โ€ฆ,cโ„“)โˆˆKโ„“ let

๐•Šโข(๐’„)=c1โข๐•Š2+โ€ฆ+cโ„“โข๐•Šโ„“+1โ€ƒandโ€ƒ๐•‹โข(๐’„)=c1โข๐•‹1+โ€ฆ+cโ„“โข๐•‹โ„“.

For aโˆˆ{2,โ€ฆ,โ„“+2} and iโˆˆ{1,โ€ฆ,โ„“} let

ba,i=(ฮธa-ฮธ1-a)โข((ฮธa-1)โข(ฮธ1-a-1))i-1,

and define the โ„“ร—โ„“-matrix B over K as

B=(b2,1b2,2โ€ฆb2,โ„“b3,1b3,2โ€ฆb3,โ„“โ‹ฎโ‹ฎโ‹ฑโ‹ฎbโ„“+1,1bโ„“+1,2โ€ฆbโ„“+1,โ„“)โˆˆKโ„“ร—โ„“.

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 fโˆˆHomPโข(TโˆงT,T).

  1. There exists a unique ๐’„โˆˆ๐’ชโ„“ with f=๐•‹โข(๐’„).

  2. There exists a unique ๐’…โˆˆKโ„“ with f=๐•Šโข(๐’…).

  3. The matrix B describes a base change from {๐•Š2,โ€ฆ,๐•Šโ„“+1} to {๐•‹1,โ€ฆ,๐•‹โ„“}, that is, ๐’„=๐’…โขB.

If ๐’„โˆˆ๐’ชโ„“, then both ๐•‹โข(๐’„) and ๐•Šโข(๐’„) lie in HomPโข(TโˆงT,T). By Lemma 4.1, {๐•‹1,โ€ฆ,๐•‹โ„“} forms an ๐’ช-basis for HomPโข(TโˆงT,T). There exists ๐’„โˆˆKโ„“โˆ–๐’ชโ„“ with ๐•Šโข(๐’„)โˆˆHomPโข(TโˆงT,T), which shows that the set {๐•Š2,โ€ฆ,๐•Šโ„“+1} generates HomPโข(TโˆงT,T), but not as an ๐’ช-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 ๐’ฎn can be obtained as Cn,eโข(๐•Šโข(๐’„)), where ๐•Šโข(๐’„) has image Tn; thus we define

ฮ“n=(๐”ญn-1)โ„“โขB-1,
ฮ”n={๐’„โˆˆKโ„“โˆฃCn,eโข(๐•Šโข(๐’„))โˆˆ๐’ฎn}
=ฮ“nโˆ–ฮ“n+1,
ฮ”n,e*={๐’„โˆˆKโ„“โˆฃCn,eโข(๐•Šโข(๐’„))โˆˆ๐’ฎnโˆ—}โŠ†ฮ”n.

Note that ฮ”n=ฮบโขฮ”n-1.

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

Lemma 4.2

For i=1,โ€ฆ,โ„“ let ๐ญ๐ข be the i-th row of B-1.

  1. We have ๐•‹i=๐•Šโข(๐’•๐’Š) and valโข(๐’•๐’Š)=-2โขi+1.

  2. If ๐’—=a1โข๐’•๐Ÿ+โ€ฆ+aโ„“โข๐’•โ„“โˆˆฮ”1 with each aiโˆˆK, then

    valโข(๐’—)=minโก{valโข(aiโข๐’•๐’Š)โˆฃi=1,โ€ฆ,โ„“}.

Proof.

(a) The proof of [11, Proposition 8.3.8] shows that detโก(B)โˆˆ๐”ญโ„“2โˆ–๐”ญโ„“2+1, hence B is invertible over K. By Lemma 4.1, the i-th row ๐’•๐’Š of B-1 satisfies ๐•‹i=๐•Šโข(๐’•๐’Š); it remains to analyse the valuation of ๐’•๐’Š. It is straightforward to see that each entry ba,i of B has valuation valโข(ba,i)=๐”ญ2โขi-1, hence

B=Uโขdiagโก(ฮบ,ฮบ3,โ€ฆ,ฮบ2โขโ„“-1),

where U is an โ„“ร—โ„“ matrix with entries in ๐’ฐ only. Since detโก(B)โˆˆ๐”ญโ„“2โˆ–๐”ญโ„“2+1, this implies that detโก(U)โˆˆ๐’ฐ. Moreover,

B-1=diagโก(ฮบ-1,ฮบ-3,โ€ฆ,ฮบ-2โขโ„“+1)โขU-1.

Since detโก(U)โˆˆ๐’ฐ, Cramerโ€™s rule for matrix inverses shows that each entry of U-1 lies in ๐’ช. Since detโก(U-1)โˆˆ๐’ฐ, each row and column of U-1 contains at least one element in ๐’ชโˆ–๐”ญ. 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 -2โขi+1.

(b) Let ฮผ=minโก{valโข(aiโข๐’•๐’Š)โˆฃi=1,โ€ฆ,โ„“}, J={iโˆˆ1,โ€ฆ,โ„“โˆฃvalโข(aiโข๐’•๐’Š)=ฮผ}. Clearly, valโข(๐’—)โ‰ฅฮผ, and we have to show equality. If |J|=1, then valโข(๐’—)=ฮผ follows readily; thus we suppose J={i1,โ€ฆ,in} with i1<โ€ฆ<in and 1<n. Note that valโข(๐’—)=ฮผ if and only if

(4.1)valโข(ai1โข๐’•๐’Š๐Ÿ+โ€ฆ+ainโข๐’•๐’Š๐’)=ฮผ.

Since valโข(aijโข๐’•๐’Š๐’‹)=ฮผ for all j, we can assume that ฮผ=0: simply replace ๐’— by a multiple ฮบ-valโข(๐’—)โข๐’—. By part (a), each valโข(๐’•๐’Š๐’‹)=-2โขij+1, so valโข(aijโข๐’•๐’Š๐’‹)=0 implies valโข(aij)=2โขij-1, that is, there is a uniquely defined cijโˆˆ{1,โ€ฆ,p-1} such that

aijโข๐’•๐’Šโ‰กcijโขฮบ2โขij-1โข๐’•๐’Šmod(๐”ญ)โ„“.

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

valโข(ai1โข๐’•๐’Š๐Ÿ+โ€ฆ+ainโข๐’•๐’Š๐’)>0.

This means that ai1โข๐’•๐’Š๐Ÿ+โ€ฆ+ainโข๐’•๐’Š๐’โˆˆ(๐”ญ)โ„“, and so

ci1โขฮบ2โขi1-1โข๐’•๐’Š๐Ÿ+โ€ฆ+cinโขฮบ2โขin-1โข๐’•๐’Š๐’โˆˆ(๐”ญ)โ„“.

In other words, if (4.1) is false, then there are c1,โ€ฆ,cโ„“โˆˆ{0,โ€ฆ,p-1}, not all 0, such that

(4.2)c1โขฮบ1โข๐’•๐Ÿ+c2โขฮบ3โข๐’•๐Ÿ+โ€ฆ+cโ„“โขฮบ2โขโ„“-1โข๐’•โ„“โˆˆ(๐”ญ)โ„“.

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 ๐’•๐Ÿ,โ€ฆ,๐’•โ„“, equation (4.2) is false if and only if the rows of the matrix

M=diagโก(ฮบ,ฮบ3,โ€ฆ,ฮบ2โขโ„“-1)โขB-1โˆˆ๐’ชโ„“ร—โ„“

are linearly independent over ๐’ช/๐”ญ, which is isomorphic to the field with p-elements. This is the case if and only if M is invertible over ๐’ช/๐”ญ, if and only if

M-1=Bโขdiagโก(ฮบ-1,ฮบ-3,โ€ฆ,ฮบ-2โขโ„“+1)โˆˆ๐’ชโ„“ร—โ„“

is invertible over ๐’ช/๐”ญ. It follows from the definition of the entries ba,i of B that

B=diagโก(x2,โ€ฆ,xโ„“+1)โขVโข(z2,โ€ฆ,zโ„“+1)โขdiagโก(ฮบ1,ฮบ3,โ€ฆ,ฮบ2โขโ„“-1),

where

xa=ฮธ1-aโข(1+ฮธ+โ€ฆ+ฮธ2โขa-2),za=(1+ฮธ+โ€ฆ+ฮธa-1)โข(1+ฮธ+โ€ฆ+ฮธ-a)โ€ƒfor all a,

and Vโข(z2,โ€ฆ,zโ„“+1) is a Vandermonde matrix with parameters z2,โ€ฆ,zโ„“+1. This yields

M-1=diagโก(x2,โ€ฆ,xโ„“+1)โขVโข(z2,โ€ฆ,zโ„“+1).

Note that each xaโ‰ก(2โขa-1modp)mod๐”ญโ‰ 0, and all zaโ‰ก(-a2+amodp)mod๐”ญโ‰ 0 are pairwise distinct. This proves that the determinant of M-1 is a unit in ๐’ช/๐”ญ; therefore M is invertible over ๐’ช/๐”ญ, which proves that (4.2) cannot be true. Thus (4.1) must hold, and the lemma is proved. โˆŽ

4.2 The isomorphism problem and automorphism groups

Recall that the groups in ๐’ฎn can be constructed as C=Cn,eโข(๐•Šโข(๐’„)) with ๐’„โˆˆฮ”n. 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 Autโข(C) can be approached using the general ideas for group extensions, see for example [17].

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

ฯa:๐’ฐโ†’๐’ฐ,uโ†ฆu-1โขฯƒaโข(u)โขฯƒ1-aโข(u),

with aโˆˆ{2,โ€ฆ,โ„“+1} play an important role. We first recall an action on ฮ”n, motivated by [10].

Lemma 4.3

The element (u,ฯƒb)โˆˆUโ‹ŠGalโข(K) acts on ๐œ=(c1,โ€ฆ,cโ„“)โˆˆฮ”n via

(u,ฯƒb)โข(๐’„)=(ฯ2โข(u)-1โขฯƒbโข(c1),โ€ฆ,ฯโ„“+1โข(u)-1โขฯƒbโข(cโ„“))โˆˆฮ”n.

This induces an action of Uโ‹ŠGalโข(K) on ฮ“n and on the set of cosets ฮ“n/ฮ“n+e for each eโˆˆN.

Proof.

We explain the origin of this action. Every automorphism ฮฒโˆˆAutโข(S) acts on fโˆˆHomPโข(TโˆงT,Tn) via fโ†ฆฮฒโข[f]=ฮฒโˆ˜fโˆ˜(ฮฒ-1โˆงฮฒ-1)|TโˆงT; if f is surjective, then so is ฮฒโข[f]. For the following, recall the notation of Lemma 3.2. If ฮฒ is in the kernel of the map ฯ€:Autโข(S)โ†’Autโข(T), then ฮฒโข[f]=f for every fโˆˆHomPโข(TโˆงT,Tn). Now let ฮฒ=ฮฑโข(u,b) with (u,ฯƒb)โˆˆ๐’ฐโ‹ŠGalโข(K). If aโˆˆ{2,โ€ฆ,โ„“+1}, cโˆˆK, and xโˆงyโˆˆTโˆงT, then a short computation shows that ฮฒโข[cโข๐•Ša]โข(xโˆงy)=ฯaโข(u-1)โขฯƒbโข(c)โข๐•Šaโข(xโˆงy). This implies that f=๐•Šโข(๐’„) is mapped to ฮฒโข[f]=๐•Šโข((u,ฯƒb)โข(๐’„)), and (u,ฯƒb)โข(๐’„)โˆˆฮ”n as required. โˆŽ

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,eโข(๐•Šโข(๐’„)) be defined as above and let

ฮป:Autโข(S)โ†’Autโข(Sn)ร—Autโข(Tn/Tn+e),
ฮปC:Autโข(C)โ†’Autโข(Sn)ร—Autโข(Tn/Tn+e)

be induced by the natural restrictions. It is easy to show that the kernel of ฮปC is isomorphic to Z1โข(Sn,Tn/Tn+e); its image is described in [17] using cohomology. Here we describe the image of ฮปC in a different way that will be more useful in our setting. Recall from Lemma 3.2 that Autโข(S) is an extension of Z1โข(P,T) by ๐’ฐโ‹ŠGalโข(K).

Theorem 4.4

Let nโ‰ฅmaxโก{m,8} and eโˆˆ{0,โ€ฆ,n-m}; let ๐œ,๐โˆˆฮ”n.

  1. The groups Cn,eโข(๐•Šโข(๐’„)) and Cn,eโข(๐•Šโข(๐’…)) are isomorphic if and only if ๐’„+ฮ“n+e and ๐’…+ฮ“n+e lie in the same orbit under the action of ๐’ฐโ‹ŠGalโข(K) as defined in Lemma 4.3.

  2. The group Autโข(Cn,eโข(๐’„)) is an extension of the p-group Z1โข(Sn,Tn/Tn+e) by ฮป(Z1(P,T).ฮฃ), where ฮฃ is the stabiliser of ๐’„+ฮ“n+e in ๐’ฐโ‹ŠGalโข(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 ๐’ฐโ‹ŠGalโข(K).

(a) Every group of depth e in โ„ฌn is an extension of Tn/Tn+e by the root Sn of โ„ฌn, 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 H2โข(Sn,Tn/Tn+e), and the isomorphism problem of such extensions can be solved by considering the action of the group of compatible pairs Compโข(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 ฮปโข(ฮฑ)=(ฮฑ|Sn,ฮฑ|Tn/Tn+e) defined by ฮฑโˆˆAutโข(S), acting on certain cohomology classes defined by surjective homomorphisms TโˆงTโ†’Tn. This action is discussed in detail in [3, Section 4.2], and it turns out that one has to consider exactly the automorphisms ฮฑโข(u,b) of S which are defined by elements (u,ฯƒb)โˆˆ๐’ฐโ‹ŠGalโข(K); the automorphisms ฮฑโข(s) with sโˆˆT act trivially. Putting all this together, the statement of the theorem follows. We note that in [4, 3] groups of depth e in โ„ฌn are described as extensions of T/Te+1 by Sn; however, Tn/Tn+eโ‰…T/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 ฮณโˆˆH2โข(Sn,Tn/Tn+e) be a cohomology class defining Cn,eโข(๐•Šโข(๐’„)). It is shown in [17] that the image of ฮปC is isomorphic to the stabiliser of ฮณ in Compโข(Sn,Tn/Tn+e), and that the kernel of ฮปC is isomorphic to the p-group Z1โข(Sn,Tn/Tn+e). The proof of [3, Lemma 5.4] now shows that the stabiliser of ฮณ in the group of compatible pairs is isomorphic to the stabiliser of ๐’„+ฮ“n+e in Autโข(Sn+e), where ฮฑโˆˆAutโข(Sn+e) acts as the compatible pair ฮปโข(ฮฑ). Note that Z1โข(P,T).ฮฃ is the stabiliser of ๐’„+ฮ“n+e in Autโข(S); since Autโข(S)โ†’Autโข(Sn+e) is surjective (and its kernel acts trivially on ๐’„+ฮ“n+e), the claim follows. โˆŽ

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 ๐’ฎnโˆ—

The automorphism group of a skeleton group C=Cn,eโข(๐’„) with ๐’„โˆˆฮ”n is described in Theorem 4.4, and we have

Aut(C)โ‰…Z1(Sn,Tn/Tn+e).ฮป(Z1(P,T).ฮฃ),

where ฮฃ is the stabiliser of ๐’„+ฮ“n+e in ๐’ฐโ‹ŠGalโข(K). Note that Z1โข(Sn,Tn/Tn+e) and ฮปโข(Z1โข(P,T)) are both p-groups, and also the subquotient ฮปCโข(๐’ฐ2โˆฉฮฃ) induces a p-group in Autโข(C). Recall that ๐’ฐ=ใ€ˆฯ‰ใ€‰ร—ใ€ˆฮธใ€‰ร—๐’ฐ2; by Lemma 4.3, the element ฮธโˆˆ๐’ฐ acts trivially in Autโข(C), and ฯ‰โˆˆ๐’ฐ acts by multiplication with ฯ‰-1. In particular, ฯ‰ cannot stabilise any non-trivial element in ฮ“n/ฮ“n+e. In summary, if e>0, then ๐’„+ฮ“n+e is non-trivial and Autโข(C) is an extension of a p-group by a subgroup of Galโข(K). Using the notation of Theorem 4.4, we have proved the following result; recall that Galโข(K) is cyclic of order d=p-1 and generated by ฯƒ.

Lemma 5.1

If nโ‰ฅmaxโก{m,8} and eโˆˆ{1,โ€ฆ,n-m}, then

ฮ”n,e*={๐’„โˆˆฮ”nโˆฃthere exists โขuโˆˆ๐’ฐโข with โข(u,ฯƒ)โข(๐’„)โ‰ก๐’„modฮ“n+e}.

It remains to determine which units uโˆˆ๐’ฐ arise in this setting, and also how to solve the isomorphism problem for the groups defined by elements in ฮ”n,e*. For this purpose, the following two sets are important; they are defined for yโˆˆโ„ค; recall that ฮ”n=ฮ“nโˆ–ฮ“n+1:

ฮฆn,y={๐’„โˆˆฮ“nโˆฃ(ฯ‰y,ฯƒ)โข(๐’„)=๐’„},
ฮ›n,y={๐’„โˆˆฮ”nโˆฃ(ฯ‰y,ฯƒ)โข(๐’„)=๐’„}.

It follows that ฮ›n,y=ฮฆn,yโˆ–ฮฆn+1,y for every yโˆˆโ„ค. Note that the elements of ฮ›n,y are โ€œglobalโ€ fixed points, while the elements of ฮ”n,e* are fixed points modulo ฮ“n+e. Every global fixed point induces fixed points modulo ฮ“n+e, hence for every eโˆˆโ„• and yโˆˆโ„ค we have

ฮ›n,yโŠ†ฮ”n,e*.

We can now state the main result of this section.

Theorem 5.2

Let ๐œโˆˆฮ”n,e* where nโ‰ฅmaxโก{m,8} and eโˆˆ{1,โ€ฆ,n-m}.

  1. There exist yโˆˆ{(n-2โขi)moddโˆฃi=1,โ€ฆ,โ„“} and ๐’‚โˆˆฮ›n,y such that Cn,eโข(๐•Šโข(๐’„))โ‰…Cn,eโข(๐•Šโข(๐’‚)).

  2. If ๐’„โˆˆฮ›n,y1 and ๐’…โˆˆฮ›n,y2 with y1,y2โˆˆโ„ค, then Cn,eโข(๐•Šโข(๐’„))โ‰…Cn,eโข(๐•Šโข(๐’…)) if and only if y1โ‰กy2modd and ๐’„=(u,1)โข(๐’…)modฮ“n+e for some uโˆˆโ„คp*.

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 5.2โ€‰(a)

Throughout this section let nโ‰ฅmaxโก{m,8} and eโˆˆ{1,โ€ฆ,n-m}.

Lemma 5.3

If ๐œ,๐โˆˆฮ”n,e*, then

Cn,eโข(๐•Šโข(๐’„))โ‰…Cn,eโข(๐•Šโข(๐’…))

if and only if (u,1)โข(๐œ)โ‰ก๐modฮ“n+e for some uโˆˆU.

Proof.

By Theorem 4.4, if Cn,eโข(๐•Šโข(๐’„))โ‰…Cn,eโข(๐•Šโข(๐’…)), then

๐’…โ‰ก(w,ฯƒb)โข(๐’„)modฮ“n+e

for some ฯƒbโˆˆGalโข(K) and wโˆˆ๐’ฐ. By Lemma 5.1, there exists an element vโˆˆ๐’ฐ with (v,1)โข(๐’„)โ‰ก(1,ฯƒ)โข(๐’„)modฮ“n+e. Write ฯƒb=ฯƒh for some hโˆˆ{1,โ€ฆ,d}, and let u=wโขvhโˆˆ๐’ฐ. Then

๐’…โ‰ก(w,ฯƒh)โข(๐’„)โ‰ก(w,1)โข(1,ฯƒh)โข(๐’„)โ‰ก(w,1)โข(vh,1)โข(๐’„)โ‰ก(u,1)โข(๐’„)modฮ“n+e,

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

Lemma 5.4

If ๐œโˆˆฮ”n,e*, then Cn,eโข(Sโข(๐œ))โ‰…Cn,eโข(Sโข(๐)) for some ๐โˆˆฮ”n,e* such that (v,ฯƒ)โข(๐)โ‰ก๐modฮ“n+e for some vโˆˆZp*.

Proof.

By Lemma 5.1, there exists uโˆˆ๐’ฐ with (u,ฯƒ)โข(๐’„)โ‰ก๐’„modฮ“n+e. We use Lemma 2.2 to decompose u=vโขx with vโˆˆkerโกฯ‡=โ„คp* and xโˆˆimโกฯ‡. Choose yโˆˆ๐’ฐ with x-1=ฯ‡โข(y)=yโขฯƒโข(y)-1, and define ๐’…=(y,1)โข(๐’„). By Theorem 4.4, we have Cn,eโข(๐’…)โ‰…Cn,eโข(๐’„), and the claim follows from

๐’…โ‰ก(y,1)โข(๐’„)
โ‰ก(y,1)โข(u,ฯƒ)โข(๐’„)
โ‰ก(y,1)โข(u,ฯƒ)โข(y-1,1)โข(๐’…)
โ‰ก(x-1โขu,ฯƒ)โข(๐’…)
โ‰ก(v,ฯƒ)โข(๐’…)modฮ“n+e.โˆŽ

In the next lemma we investigate the set ฮ›n,y=ฮฆn,yโˆฉฮ”n in more detail. Recall that ฯ‰ acts by multiplication by ฯ‰-1 on ฮ“n, and therefore

(5.1)ฮฆn,y={๐’„โˆˆฮ“nโˆฃ(1,ฯƒ)โข(๐’„)=ฯ‰yโข๐’„}.
Lemma 5.5

For each y, the set ฮฆn,y is a Zp-sublattice of ฮ“n; moreover

ฮ“n=ฮฆn,0โŠ•โ€ฆโŠ•ฮฆn,d-1.

Proof.

The action of ๐’ฐโ‹ŠGalโข(K) on ฮ“n defined in Lemma 4.3 extends to an action on the โ„“-fold direct product Kโ„“; in particular, (1,ฯƒ) acts via

(x1,โ€ฆ,xโ„“)โ†ฆ(ฯƒโข(x1),โ€ฆ,ฯƒโข(xโ„“)).

By Lemma 2.1, ฯƒ:Kโ†’K is diagonalisable with eigenvalues ฯ‰0,โ€ฆ,ฯ‰d-1, and each eigenspace has dimension 1. Thus, the action of (1,ฯƒ) on Kโ„“ is diagonalisable with eigenvalues ฯ‰0,โ€ฆ,ฯ‰d-1 and each eigenspace has dimension โ„“. We denote by Vi the eigenspace with eigenvalue ฯ‰i in Kโ„“.

It follows from (5.1) that ฮฆn,y=Vyโˆฉฮ“n, hence ฮฆn,0โŠ•โ€ฆโŠ•ฮฆn,d-1โ‰คฮ“n and each ฮฆn,y is a sublattice of ฮ“n. It remains to show the following inclusion: ฮ“nโŠ†ฮฆn,0โŠ•โ€ฆโŠ•ฮฆn,d-1. For this purpose, consider wโˆˆฮ“n and write w=w0+โ€ฆ+wd-1 with wiโˆˆVi for i=1,โ€ฆ,โ„“. Let k be the number of non-zero summands wi in w. We prove by induction on k that each wiโˆˆฮฆn,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 wjโ‰ 0. It follows from Lemma 4.3 that both (1,ฯƒ)โข(w) and (ฯ‰-j,1)โข(w)=ฯ‰jโขw lie in ฮ“n, hence also their difference u=(1,ฯƒ)โข(w)-ฯ‰jโขw lies in ฮ“n. Note that u=u0+โ€ฆ+ud-1, where each ui=(ฯ‰i-ฯ‰j)โขwiโˆˆVi. By construction, u has at most k-1 non-zero summands ui and, by the induction hypothesis, we have uiโˆˆฮฆn,i for each i. Recall that โ„คp*=ใ€ˆฯ‰ใ€‰โข(1+pโขโ„คp); this implies that ฯ‰i-ฯ‰jโ‰ข0modp and hence ฯ‰i-ฯ‰jโˆˆโ„คp* for all iโ‰ j. Since ui=(ฯ‰i-ฯ‰j)โขwiโˆˆฮฆn,i, it follows that wiโˆˆฮฆn,i for all iโ‰ j. In particular, we have w-wj=w0+โ€ฆ+wj-1+wj+1+โ€ฆ+wd-1โˆˆฮ“n, and so wj=w-(w-wj)โˆˆฮ“n as well. Now clearly wjโˆˆฮฆn,j, which completes the proof. โˆŽ

The next lemma considers ๐’…โˆˆฮ”n,e* and vโˆˆโ„คp* with (v,ฯƒ)โข(๐’…)โ‰ก๐’…modฮ“n,e, as in Lemma 5.4. Recall that every vโˆˆโ„คp* can be written as

v=ฯ‰uโข(1+pโขx)=ฯ‰-yโข(1+pโขx)

for some u,yโˆˆ{0,โ€ฆ,d-1} and xโˆˆโ„คp.

Lemma 5.6

Let ๐โˆˆฮ”n,e* and vโˆˆZp* with (v,ฯƒ)โข(๐)โ‰ก๐modฮ“n,e, and write v=ฯ‰-yโข(1+pโขx) for some yโˆˆ{0,โ€ฆ,d-1} and xโˆˆZp. There exists ๐šโˆˆฮ›n,y with Cn,eโข(Sโข(๐))=Cn,eโข(Sโข(๐š)).

Proof.

By definition, ๐’…โˆˆฮ”n=ฮ“nโˆ–ฮ“n+1; by Lemma 5.5, we can decompose ๐’…=๐’…0+โ€ฆ+๐’…d-1 with each ๐’…iโˆˆฮฆn,i. As (v,ฯƒ)โข(๐’…)โ‰ก๐’…modฮ“n+e and vโˆˆโ„คpโˆ—, it follows that

(v,ฯƒ)โข(๐’…)-๐’…=(v-1โขฯ‰0-1)โข๐’…0+โ€ฆ+(v-1โขฯ‰d-1-1)โข๐’…d-1โˆˆฮ“n+e.

As v-1โขฯ‰i-1โˆˆโ„คp, Lemma 5.5 yields that (v-1โขฯ‰i-1)โข๐’…iโˆˆฮ“n+e for each i. Using v=ฯ‰-yโข(1+pโขx), it follows that v-1โขฯ‰i-1โ‰กฯ‰i-y-1modp; in particular, v-1โขฯ‰i-1โˆˆโ„คp* and ๐’…iโˆˆฮ“n+e for all iโ‰ y. We can now choose ๐’‚=๐’…yโˆˆฮฆn,y; we have shown that ๐’…โ‰ก๐’‚modฮ“n+e, thus

Cn,eโข(๐•Šโข(๐’…))=Cn,eโข(๐•Šโข(๐’‚)).

As ๐’…โˆˆฮ”n, it follows that ๐’‚โˆˆฮ”n. Hence ๐’‚โˆˆฮ›n,y as claimed. โˆŽ

Lemma 5.7

We have ฮ›n,yโ‰ โˆ… if and only if (1,ฯƒ) has an eigenvector in ฮ”1 with eigenvalue ฯ‰y-n+1.

Proof.

Let ๐’“โˆˆฮ”1 be an eigenvector of (1,ฯƒ) with eigenvalue ฯ‰y-n+1. By Lemma 2.1, there exists an eigenvector xโˆˆ๐”ญn-1โˆ–๐”ญn of ฯƒ with eigenvalue ฯ‰n-1. Then xโข๐’“โˆˆฮ”n and

(ฯ‰y,ฯƒ)โข(xโข๐’“)=ฯ‰-yโขฯƒโข(x)โข(1,ฯƒ)โข(๐’“)
=ฯ‰-yโขฯ‰n-1โขxโขฯ‰y-n+1โข๐’“
=xโข๐’“,

so xโข๐’“โˆˆฮ›n,y. Conversely, let ๐’„โˆˆฮ›n,y. Then ๐’„ is an eigenvector of (1,ฯƒ) with eigenvalue ฯ‰y. If xโˆˆ๐”ญ-n+1โˆ–๐”ญ-n+2 is an eigenvector of ฯƒ with eigenvalue ฯ‰-n+1, then xโข๐’„โˆˆฮ”1 and

(1,ฯƒ)โข(xโข๐’„)=ฯ‰-n+1โขxโข(1,ฯƒ)โข(๐’„)
=ฯ‰-n+1โขxโขฯ‰yโข๐’„
=ฯ‰y-n+1โขxโข๐’„.โˆŽ

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

Lemma 5.8

The eigenvalues of (1,ฯƒ) on ฮ”1 are {ฯ‰-2โขi+1โˆฃi=1,โ€ฆ,โ„“}.

Proof.

The main tool in the following proof is Lemma 4.2: with Lemma 2.1 it implies that the eigenvalue of an eigenvector ๐’‡โˆˆฮ”1 of (1,ฯƒ) is ฯ‰valโข(๐’•), where valโข(๐’•) is the valuation defined in Section 2. The eigenvalues on ฮ”1=ฮ“1โˆ–ฮ“2 coincide with the eigenvalues of (1,ฯƒ) on the elementary abelian quotient ฮ“1/ฮ“2 of rank โ„“; in particular, this number of different eigenvalues is at most โ„“. We show that it is exactly โ„“. To prove this, we use the notation of Lemma 4.2 and define an isomorphism

ฯˆ:Cpโ„“โ‰…ฮ“1/ฮ“2,(a1,โ€ฆ,aโ„“)โ†ฆa1โข๐’•๐Ÿ+โ€ฆ+aโ„“โข๐’•โ„“+ฮ“2,

recall that each ๐’•๐’‹ has valuation valโข(๐’•๐’‹)=-2โขj+1. In the following we let (1,ฯƒ) act on Cpโ„“ via ฯˆ; in particular, the action of (1,ฯƒ) on Cpโ„“ is diagonalisable. Our claim is that for each iโˆˆ{1,โ€ฆ,โ„“} there is, up to scalar multiples, a unique eigenvector (a1,โ€ฆ,ai,0,โ€ฆ,0)โˆˆCpโ„“ of (1,ฯƒ) with aiโ‰ 0, and that this eigenvector has eigenvalue ฯ‰-2โขi+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=โ„“. Since the action of (1,ฯƒ) on Cpโ„“ is diagonalisable, there must be an eigenvector a=(a1,โ€ฆ,aโ„“)โˆˆCpโ„“ with aโ„“โ‰ 0. This eigenvector comes from an eigenvector ๐’‡=a1โข๐’•๐Ÿ+โ€ฆ+aโ„“โข๐’•โ„“+๐’ˆโˆˆฮ”1 for some ๐’ˆโˆˆฮ”2. Recall that ฮ”2=ฮบโขฮ”1โŠ†(๐”ญ-2โขโ„“+2)โ„“. Now

valโข(aโ„“โข๐’•โ„“)=valโข(aโ„“)+valโข(๐’•โ„“)=valโข(๐’•โ„“)=-2โขโ„“+1

and valโข(๐’ˆ)โ‰ฅ-2โขโ„“+2 imply that valโข(๐’‡)=-2โขโ„“+1, see Lemma 4.2, whence the eigenvalue of ๐’‡ (and a) is ฯ‰-2โขโ„“+1. For the uniqueness, consider an eigenvector aโ€ฒ=(a1โ€ฒ,โ€ฆ,aโ„“โ€ฒ)โˆˆCpโ„“ with aโ„“โ€ฒโ‰ 0; as just shown, the eigenvalue is ฯ‰-2โขโ„“+1. Without loss of generality, we can assume that aโ„“=aโ„“โ€ฒ. Suppose, for a contradiction, that a and aโ€ฒ are linearly independent, so that

b=a-aโ€ฒ=(b1,โ€ฆ,bโ„“-1,0)โˆˆCpโ„“

is an eigenvector of (1,ฯƒ) with eigenvalue ฯ‰-2โขโ„“+1. This eigenvector comes from an eigenvector ๐’–=b1โข๐’•๐Ÿ+โ€ฆ+bโ„“-1โข๐’•โ„“-๐Ÿ+๐’‰โˆˆฮ”1 for some ๐’‰โˆˆฮ”2. But now valโข(๐’–)โ‰ฅ-2โขโ„“+2, so the eigenvalue of b cannot be ฯ‰-2โขโ„“+1, a contradiction. (Indeed, if ฯ‰valโข(๐’–)=ฯ‰-2โขโ„“+1, then valโข(๐’–)โ‰ฅ-2โขโ„“+2 forces valโข(๐’–)โ‰ฅ-2โขโ„“+p>0, so ๐’–โˆˆฮ”2 and (b1,โ€ฆ,bโ„“-1,0)=(0,โ€ฆ,0), which is not possible.) This proves that there is, up to scalar multiples, a unique eigenvector of (1,ฯƒ) in Cpโ„“ of the form (a1,โ€ฆ,aโ„“) with aโ„“โ‰ 0, and that the corresponding eigenvalue is ฯ‰-2โขโ„“+1.

Induction hypothesis. Our induction hypothesis now is that for each index jโˆˆ{i+1,โ€ฆ,โ„“} there is, up to scalar multiples, a unique eigenvector of (1,ฯƒ) in Cpโ„“ of the form

vj=(bj,1,โ€ฆ,bj,j,0,โ€ฆ,0)โˆˆCpโ„“

with bj,jโ‰ 0, and that the corresponding eigenvalue is ฯ‰-2โขj+1. Note that vj comes from an eigenvector

๐’˜๐’‹=bj,1โข๐’•๐Ÿ+โ€ฆ+bj,jโข๐’•๐’‹+๐’‰๐’‹โˆˆฮ”1

for some ๐’‰๐’‹โˆˆฮ”2, with

valโข(๐’‰๐’‹)โ‰ฅvalโข(bj,1โข๐’•๐Ÿ+โ€ฆ+bj,jโข๐’•๐’‹)=valโข(bj,jโข๐’•๐’‹)=-2โขj+1.

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

a=(a1,โ€ฆ,ai,0,โ€ฆ,0)โˆˆCpโ„“

of (1,ฯƒ) with aiโ‰ 0: if not, then there would be a basis of Cpโ„“ consisting of {vi+1,โ€ฆ,vโ„“} and i additional eigenvectors each having 0 as k-th entry for all k=i,โ€ฆ,โ„“; 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 ฯ‰-2โขi+1. Note that a comes from an eigenvector

๐’‡=a1โข๐’•๐Ÿ+โ€ฆ+aiโข๐’•๐’Š+๐’ˆโˆˆฮ”1

for some ๐’ˆโˆˆฮ”2; in the following, write r=valโข(๐’ˆ). If

rโ‰ฅvalโข(a1โข๐’•๐Ÿ+โ€ฆ+aiโข๐’•๐’Š)=valโข(aiโข๐’•๐’Š)=valโข(ai)+valโข(aiโข๐’•๐’Š)=valโข(aiโข๐’•๐’Š)=-2โขi+1,

then valโข(๐’‡)=-2โขi+1 by Lemma 4.2, and it follows that the eigenvalue is ฯ‰-2โขi+1. It remains to consider the case r<-2โขi+1, so that the eigenvalue of a is ฯ‰r. We show that this is not possible; we achieve this by modifying ๐’ˆ by our known eigenvectors ๐’˜๐’Š+๐Ÿ,โ€ฆ,๐’˜โ„“ until we obtain a contradiction. For this purpose, write

๐’ˆ=u1โข๐’•๐Ÿ+โ€ฆ+uโ„“โข๐’•โ„“โˆˆฮ”2

and let sโˆˆ{1,โ€ฆ,โ„“} be minimal with valโข(usโข๐’•๐’”)=valโข(๐’ˆ)=r; such an s exists by Lemma 4.2. Note that r=valโข(usโข๐’•๐’”)>-2โขs+1 since usโˆˆ๐”ญ, hence r+2โขs-1>0. Since r<-2โขi+1 by assumption, this implies sโˆˆ{i+1,โ€ฆ,โ„“}. By the induction hypothesis, we know the existence of the eigenvector

๐’˜๐’”=bs,1โข๐’•๐Ÿ+โ€ฆ+bs,sโข๐’•๐’”+๐’‰๐’”โˆˆฮ”1

of (1,ฯƒ) with eigenvalue ฯ‰-2โขs+1. Let kโˆˆ๐”ญr+2โขs-1โˆ–๐”ญr+2โขs be an eigenvector of ฯƒ with eigenvalue ฯ‰r+2โขs-1. Now ๐’‡โ€ฒ=๐’‡-kโข๐’˜๐’” is an eigenvector of (1,ฯƒ) with eigenvalue ฯ‰r, and that both ๐’‡ and ๐’‡โ€ฒ correspond to

a=(a1,โ€ฆ,ai,0,โ€ฆ,0)โˆˆCpโ„“

since kโˆˆ๐”ญ. In particular,

๐’‡โ€ฒ=a1โข๐’•๐Ÿ+โ€ฆ+aiโข๐’•๐’Š+(๐’ˆ-kโข๐’˜๐’”)
=a1โข๐’•๐Ÿ+โ‹ฏ+aiโข๐’•๐’Š
+((u1-kโขbs,1)โข๐’•๐Ÿ+โ‹ฏ+(us-kโขbs,s)โข๐’•๐’”+us+1โข๐’•๐’”+๐Ÿ+โ‹ฏ+uโ„“โข๐’•โ„“)-kโข๐’‰๐’”โŸ=โฃ:๐’ˆโ€ฒ.

Since both us,kโขbs,sโˆˆ๐”ญr-2โขs+1โˆ–๐”ญr-2โขs+2, we can replace k by a suitable scalar multiple of k such that us-kโขbs,sโˆˆ๐”ญr-2โขs+2, and so valโข((us-kโขbs,s)โข๐’•๐’”)>r; note that valโข((uj-kโขbs,j)โข๐’•๐’‹)โ‰ฅvalโข(ujโข๐’•๐’‹) for all j. In conclusion, we have found ๐’‡โ€ฒ=a1โข๐’•๐Ÿ+โ€ฆ+aiโข๐’•๐’Š+๐’ˆโ€ฒ with ๐’ˆโ€ฒโˆˆฮ”2 such that if ๐’ˆโ€ฒ=u1โ€ฒโข๐’•๐Ÿ+โ€ฆ+uโ„“โ€ฒโข๐’•โ„“ and sโ€ฒโˆˆ{1,โ€ฆ,โ„“} is minimal with valโข(usโ€ฒโข๐’•๐’”โ€ฒ)=valโข(๐’ˆโ€ฒ)=valโข(๐’ˆ)=r, then sโ€ฒ>s. (Note that ๐’‡โ€ฒ has eigenvalue ฯ‰r and r<-2โขi+1, so we must indeed have valโข(๐’ˆโ€ฒ)=r=valโข(๐’ˆ).) Now we iterate this argument until we find an eigenvector

๐’‡^=a1โข๐’•๐Ÿ+โ€ฆ+aiโข๐’•๐’Š+๐’ˆ^

of (1,ฯƒ) with eigenvalue ฯ‰r and ๐’ˆ^โˆˆฮ”2 such that if ๐’ˆ^=u1^โข๐’•๐Ÿ+โ€ฆ+uโ„“^โข๐’•โ„“, then valโข(uj^โข๐’•๐’‹)>r for all j. But then Lemma 4.2 implies that valโข(๐’ˆ^)>r, a contradiction to ฯ‰valโข(f)=ฯ‰r and valโข(๐’ˆ^)=r. In summary, this proves rโ‰ฅ-2โขi+1, hence the eigenvalue of an eigenvector a=(a1,โ€ฆ,ai,0,โ€ฆ,0) with aiโ‰ 0 must be ฯ‰-2โขi+1.

Uniqueness. Consider a second eigenvector aโ€ฒ=(a1โ€ฒ,โ€ฆ,aiโ€ฒ,0,โ€ฆ,0)โˆˆCpโ„“ of (1,ฯƒ) with aiโ€ฒโ‰ 0; as proved in the previous paragraph, the eigenvalue of aโ€ฒ is ฯ‰-2โขi+1. We claim that aโ€ฒ and a are linearly dependent. Note that a comes from an eigenvector ๐’‡=a1โข๐’•๐Ÿ+โ€ฆ+aiโข๐’•๐’Š+๐’ˆโˆˆฮ”1 for some ๐’ˆโˆˆฮ”2 with valโข(๐’ˆ)โ‰ฅ-2โขi+1. Similarly, aโ€ฒ comes from an eigenvector ๐’‡โ€ฒ=a1โ€ฒโข๐’•๐Ÿ+โ€ฆ+aiโ€ฒโข๐’•๐’Š+๐’ˆโ€ฒโˆˆฮ”1 for some ๐’ˆโ€ฒโˆˆฮ”2 with valโข(๐’ˆโ€ฒ)โ‰ฅ-2โขi+1. Suppose, for a contradiction, that a and aโ€ฒ are linearly independent. Replacing aโ€ฒ by a suitable scalar multiple, we can assume that ai=aiโ€ฒ, so that b=a-aโ€ฒ=(b1,โ€ฆ,bj,0,โ€ฆ,0), with bjโ‰ 0 and j<i, is also an eigenvector of (1,ฯƒ) with eigenvalue ฯ‰-2โขi+1. This eigenvector comes from ๐’‡^=๐’‡-๐’‡โ€ฒ=b1โข๐’•๐Ÿ+โ€ฆ+bjโข๐’•๐’‹+๐’ˆ^, where ๐’ˆ^=๐’ˆ-๐’ˆโ€ฒโˆˆฮ”2 satisfies valโข(๐’ˆ^)โ‰ฅ-2โขi+1; in fact, we must have valโข(๐’ˆ^)=-2โขi+1 since otherwise the eigenvalue of b cannot be ฯ‰-2โขi+1. Note that for all u=i+1,โ€ฆ,โ„“ we already found eigenvectors ๐’˜๐’– with eigenvalue ฯ‰-2โขu+1, thus we can use the same construction as in the previous paragraph to obtain from ๐’‡^ and ๐’˜๐’Š+๐Ÿ,โ€ฆ,๐’˜โ„“ an eigenvector ๐’‡~=b1โข๐’•๐Ÿ+โ€ฆ+bjโข๐’•๐’‹+๐’ˆ~ with eigenvalue ฯ‰-2โขi+1, where ๐’ˆ~โˆˆฮ”2 satisfies valโข(๐’ˆ~)>-2โขi+1: this is not possible since ๐’‡~ has eigenvalue ฯ‰-2โขi+1, that is, -2โขi+1=valโข(๐’‡~)=minโก{-2โขj+1,valโข(๐’ˆ~)}, but we have deduced that minโก{-2โขj+1,valโข(๐’ˆ~)}>-2โขi+1. This contradiction proves that a and aโ€ฒ must be linearly dependent. In conclusion, we have proved that, up to scalar multiples, there is a unique eigenvector of (1,ฯƒ) of the form (a1,โ€ฆ,ai,0,โ€ฆ,0) with aiโ‰ 0, and that the corresponding eigenvalue is ฯ‰-2โขi+1. This completes the induction step. โˆŽ

Lemmas 5.7 and 5.8 yield the following corollary.

Corollary 5.9

We have ฮ›n,yโ‰ โˆ… if and only if yโ‰กn-2โขimodd for some i=1,โ€ฆ,โ„“.

We can now prove Theorem 5.2โ€‰(a).

Proof of Theorem 5.2โ€‰(a).

Let ๐’„โˆˆฮ”n,e*. By Lemmas 5.4 and 5.6, there exist yโˆˆ{0,โ€ฆ,d-1} and ๐’‚โˆˆฮ›n,y such that

Cn,eโข(๐•Šโข(๐’„))โ‰…Cn,eโข(๐•Šโข(๐’‚)).

Since ฮ›n,yโ‰ โˆ…, it follows from Corollary 5.9 that yโ‰กn-2โขimodd for some iโˆˆ{1,โ€ฆ,โ„“}, as claimed. โˆŽ

5.2 Proof of Theorem 5.2โ€‰(b)

Again, we assume that nโ‰ฅmaxโก{m,8} and eโˆˆ{1,โ€ฆ,n-m}. Recall the map ฯ‡:๐’ฐโ†’๐’ฐ, uโ†ฆuโขฯƒโข(u)-1, which is discussed in Lemma 2.2.

Lemma 5.10

Let ๐œโˆˆฮ›n,y1 and ๐โˆˆฮ›n,y2 with y1,y2โˆˆZ. If Cn,eโข(Sโข(๐œ))โ‰…Cn,eโข(Sโข(๐)), then y1=y2; if uโˆˆU with ๐โ‰ก(u,1)โข(๐œ)modฮ“n+e as in Lemma 5.3, then ๐œ and ๐ are fixed points of (ฯ‡โข(u),1) modulo ฮ“n+e.

Proof.

It is shown in Lemma 5.3 that ๐’…โ‰ก(u,1)โข(๐’„)modฮ“n+e for some uโˆˆ๐’ฐ. This implies that

๐’…โ‰ก(ฯ‰y2,ฯƒ)โข(๐’…)
โ‰ก(ฯ‰y2,ฯƒ)โข(u,1)โข(๐’„)
โ‰ก(ฯ‰y2,ฯƒ)โข(u,1)โข(ฯ‰y1,ฯƒ)-1โข(๐’„)
โ‰ก(ฯ‰y2,ฯƒ)โข(u,1)โข(ฯ‰y1,ฯƒ)-1โข(u,1)-1โข(๐’…)
โ‰ก(ฯ‰y2-y1โขฯƒโข(u)โขu-1,1)โข(๐’…)
โ‰ก(ฯ‰y2-y1โขฯ‡โข(u)-1,1)โข(๐’…)modฮ“n+e.

By Theorem 4.4, the element (ฯ‰y2-y1โขฯ‡โข(u)-1,1) yields an element of the group Autโข(Cn,eโข(๐’…)), and the discussion of the automorphism groups of skeleton groups (in the beginning of Section 5) forces that y2โ‰กy1modd. Since we have y1,y2โˆˆ{0,โ€ฆ,d-1}, this yields y1=y2. In turn, this implies that ๐’… (and, by duality also ๐’„) are fixed points of (ฯ‡โข(u)-1,1) modulo ฮ“n+e. โˆŽ

Lemma 5.11

Let ๐œ,๐โˆˆฮ›n,y with yโˆˆZ. If Cn,eโข(Sโข(๐œ))โ‰…Cn,eโข(Sโข(๐)), then ๐œโ‰ก(s,1)โข(๐)modฮ“n+e for some sโˆˆZp*.

Proof.

Recall that ๐’„,๐’…โˆˆฮ›n,yโŠ†ฮ”n,e*. Thus, by Lemma 5.3, there exists uโˆˆ๐’ฐ with (u,1)โข(๐’„)โ‰ก๐’…modฮ“n+e. By Lemma 5.10, both ๐’„ and ๐’… are fixed points under (ฯ‡โข(u),1). It follows from Lemma 2.2 that the restriction ฯ‡|imโกฯ‡ induces an automorphism of imโกฯ‡; write u=sโขw with sโˆˆkerโกฯ‡=โ„คp* and wโˆˆimโกฯ‡. We want to show that (w,1)โข(๐’…)โ‰ก๐’…modฮ“n+e since this implies that

๐’„โ‰ก(u,1)โข(๐’…)โ‰ก(s,1)โข(w,1)โข(๐’…)โ‰ก(s,1)โข(๐’…)modฮ“n+e,

which proves the lemma.

We start with a more general observation: Let vโˆˆ๐’ฐ and suppose (v,1) maps the fixed point ๐’… of (ฯ‰y,ฯƒ) to another fixed point (v,1)โข(๐’…) of (ฯ‰y,ฯƒ) modulo ฮ“n+e. Then it follows from

(v,1)โข(๐’…)โ‰ก(ฯ‰y,ฯƒ)โข(v,1)โข(๐’…)โ‰ก(ฯƒโข(v),1)โข(ฯ‰y,ฯƒ)โข(๐’…)โ‰ก(ฯƒโข(v),1)โข(๐’…)modฮ“n+e

that such a (v,1) has an image (ฯ‡โข(v),1) which acts trivially on ๐’… modulo ฮ“n,e, and by duality also on ๐’„ modulo ฮ“n,e.

Since (u,1) maps the fixed point ๐’… to the fixed point ๐’„, it follows that (ฯ‡โข(u),1) stabilises ๐’… modulo ฮ“n+e. Note that ฯ‡โข(u)=ฯ‡โข(w), hence also (ฯ‡โข(w),1) stabilises ๐’… modulo ฮ“n+e. Now we iterate this argument: since (ฯ‡โข(w),1) maps the fixed point ๐’… of (ฯ‰y,1) to the fixed point ๐’… of (ฯ‰y,1), modulo ฮ“n+e, it follows from the general observation that (ฯ‡2โข(w),1) stabilises ๐’… modulo ฮ“n+e. By induction, (ฯ‡iโข(w),1) stabilises ๐’… modulo ฮ“n+e for every iโ‰ฅ1.

There is jโ‰ฅ2 such that ๐’ฐj=1+๐”ญj acts trivially on ฮ“n/ฮ“n+e: for example, choose j large enough such that ๐”ญjโ‰คpxโข๐’ช for some x with pxโขฮ“nโ‰คฮ“n+e. It follows from the definition that ฯ‡ stabilises ๐’ฐj. Since ๐’ฐjโ‰ค๐’ฐ2, the proof of Lemma 2.2 shows that ๐’ฐj=kerโก(ฯ‡|๐’ฐj)ร—imโก(ฯ‡|๐’ฐj); this implies that ฯ‡ induces an automorphism of J=(imโกฯ‡)/(imโกฯ‡|๐’ฐj), which we denote by ฯˆ. Since J is a finite group, it follows that ฯˆ has finite order, say t.

Recall that w as above lies in imโกฯ‡. If w lies in ๐’ฐj, then (w,1) acts trivially on ๐’… modulo ฮ“n+e, and there is nothing to show. If wโˆ‰๐’ฐj, then its coset c=wโขimโก(ฯ‡|๐’ฐj) in J is non-trivial, and ฯˆtโข(c)=c follows. But this means that ฯ‡tโข(w)=wโขr for some rโˆˆ๐’ฐj. As shown above, ฯ‡tโข(w)=wโขr stabilises ๐’… modulo ฮ“n+e. Since rโˆˆ๐’ฐj acts trivially on ฮ“n/ฮ“n+e, it follows that w stabilises ๐’… modulo ฮ“n+e. โˆŽ

We can now prove Theorem 5.2โ€‰(b).

Proof of Theorem 5.2(b).

Let ๐’„โˆˆฮ›n,y1 and ๐’…โˆˆฮ›n,y2. If there exists an element uโˆˆโ„คp* with ๐’„โ‰ก(u,1)โข(๐’…)modฮ“n+e, then we have Cn,eโข(๐•Šโข(๐’„))โ‰…Cn,eโข(๐•Šโข(๐’…)) by Lemma 5.3. For the converse, suppose that Cn,eโข(๐•Šโข(๐’„))โ‰…Cn,eโข(๐•Šโข(๐’…)). Then y1โ‰กy2modd follows from Lemma 5.10, and ๐’„=(u,1)โข(๐’…)modฮ“n+e for some uโˆˆโ„คp* follows from Lemma 5.11. โˆŽ

6 The ramification levels in the skeleton ๐’ฎnโˆ—

We apply Theorem 5.2 to prove the explicit description of the skeleton ๐’ฎnโˆ— as suggested in Theorem 1.1. First we consider the groups of depth 1.

Theorem 6.1

Let nโ‰ฅmaxโก{8,m}. There are โ„“ groups Gn,1,โ€ฆ,Gn,โ„“ at depth 1 in Snโˆ—; these are obtained as Gn,i=Cn,1โข(Sโข(๐œi)), where ๐œi is a fixed point of (ฯ‰ji,ฯƒ) with ji=n-2โขimodd.

Proof.

It follows from Theorem 5.2โ€‰(a) that the groups at depth 1 in ๐’ฎnโˆ— can be constructed as Cn,1โข(๐’„) with ๐’„โˆˆฮ›n,y for y=0,โ€ฆ,d-1. It is shown in Corollary 5.9 that ฮ›n,yโ‰ โˆ… if and only if y=yi=n-2โขimodd for some index iโˆˆ{1,โ€ฆ,โ„“}; this allows us to construct โ„“ groups Gn,i=Cn,1โข(๐’„๐’Š), where ๐’„๐’Šโˆˆฮ›n,yi for i=1,โ€ฆ,โ„“. Theorem 5.2โ€‰(b) shows that Gn,iโ‰…Gn,j if and only if i=j. It follows from Lemmas 5.7 and 5.8 that the eigenspace of each (ฯ‰yi,ฯƒ) in ฮ“n/ฮ“n+1 has dimension 1. Thus the elements in ฮ›n,yiโ‰ โˆ… admit exactly one isomorphism type of skeleton group for each i=1,โ€ฆ,โ„“. This completes the proof. โˆŽ

Now we consider the remaining part of the skeleton.

Theorem 6.2

Let iโˆˆ{1,โ€ฆ,โ„“} and let H be a descendant of Gn,i at depth eโˆˆ{1,โ€ฆ,n-m-1} in Snโˆ—, where Gn,i is as in Theorem 6.1. The group H has p immediate descendants in Snโˆ— if and only if emoddโˆˆ{2,4,โ€ฆ,d-2}โˆ–{d-2โขi}; otherwise, H has one immediate descendant in Snโˆ—.

Proof.

We investigate the descendants of Gn,i for a fixed iโˆˆ{1,โ€ฆ,โ„“}. By Theorems 5.2 and 6.1, our aim is to determine the orbits and stabilisers of โ„คp* acting on

ฮ n,j,e={๐’„+ฮ“n+eโˆฃ๐’„โˆˆฮ›n,j},

where j=n-2โขimodd. In the following, let ๐’ž={0,2,โ€ฆ,d-2}โˆ–{d-2โขi}.

We first consider stabilisers; recall that โ„คpโˆ—=ใ€ˆฯ‰ใ€‰โข(1+pโขโ„คp), and note that uโˆˆโ„คp* acts on ๐’…โˆˆฮ›n,j via

(u,1)โข(๐’…)=u-1โข๐’…modฮ“n+e.

Thus (u,1)โข(๐’…)โ‰ก๐’…modฮ“n+e if and only if uโˆˆ1+peยฏ with eยฏ=โŒˆe/dโŒ‰; this implies that Stabโ„คp*โข(๐’…+ฮ“n+e)=1+pโŒˆe/dโŒ‰โขโ„คp. Note that the stabiliser depends on e, but not on ๐’…. In particular, it follows that each โ„คp*-orbit has the same size, namely dโขpeยฏ-1.

Next we show that

|ฮ n,j,e|={|ฮ n,j,e+1|if emoddโˆ‰๐’ž,pโข|ฮ n,j,e+1|otherwise.

To prove this, note that |ฮ n,j,e|<|ฮ n,j,e+1| if and only if there exist ๐’…,๐’†โˆˆฮ›n,j with ๐’…โ‰ก๐’†modฮ“n+e and ๐’…โ‰ข๐’†modฮ“n+e+1. This holds if and only if there exists a fixed point ๐’“โˆˆฮ“n+eโˆ–ฮ“n+e+1 of (ฯ‰j,ฯƒ) with ๐’…=๐’„+๐’“. By Corollary 5.9, such a fixed point exists if and only if jโ‰กn+e-2โขiโ€ฒmodd for some iโ€ฒโˆˆ{1,โ€ฆ,โ„“}. Since jโ‰กn-2โขimodd, such an iโ€ฒ exists if and only if eโ‰ก2โข(iโ€ฒ-i)modd. A straightforward computation shows that

๐’ž={2โข(iโ€ฒ-i)moddโˆฃiโ€ฒ=1,โ€ฆ,โ„“}.

Thus, in summary, |ฮ n,j,e|>|ฮ n,j,e+1| if and only if emoddโˆˆ๐’ž; in this case, |ฮ n,j,e|=pโข|ฮ n,j,e+1| follows since the eigenspace of (ฯ‰j,ฯƒ) on ฮ“n+e/ฮ“n+e+1 has dimension 1.

It remains to consider eโ‰ก0modd. 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 ๐’ฎnโˆ—. If emoddโˆˆ๐’žโˆ–{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 ๐’ฎnโˆ—. โˆŽ


Communicated by Evgenii I. Khukhro


Funding statement: This research was supported by a Go8-DAAD Joint Research Co-operation Scheme, project โ€œGroups of Prime-Power Order and Coclass Theoryโ€. 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 ๐’ขโข(p,r) associated with the p-groups of coclass r. As for maximal class, the vertices of ๐’ขโข(p,r) are identified with isomorphism type representatives of the considered groups, and there is an edge Gโ†’H if and only if G is isomorphic to H/ฮณโข(H), where ฮณโข(H) is the last non-trivial term of the lower central series of H. It is known that ๐’ขโข(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 ๐’ฏ which has a unique infinite path G1โ†’G2โ†’โ‹ฏ starting at its root. The branchโ„ฌn of ๐’ฏ is the subtree of ๐’ฏ 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 ๐’ขโข(p,r). The aim of this appendix is to provide more details on known periodicity results for ๐’ขโข(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 Blackburnโ€™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 Aโ€“E, where Conjecture A is the strongest since it implies Conjectures Bโ€“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.โ€‰265]. We remark that it is Conjecture D which implies that each coclass graph ๐’ขโข(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, ๐’ฏ denotes a coclass tree with branches โ„ฌn in some specified coclass graph ๐’ขโข(p,r). For an integer k>0 let โ„ฌnโข(k) be the pruned subtree of โ„ฌn generated by all groups at depth at most k in โ„ฌn.

For pโˆˆ{2,3}, Blackburn proved that the branches of the coclass tree in ๐’ขโข(p,1) satisfy โ„ฌnโ‰…โ„ฌn+p-1 for all large enough n, that is, ๐’ขโข(p,1) is virtually periodic.

Newman [14], and later Dietrich, Eick and Feichtenschlager [5], did extensive computer experiments for ๐’ขโข(5,1), which suggest that โ„ฌnโข(n-1)โ‰…โ„ฌn+4โข(n-1) and โ„ฌn+4โˆ–โ„ฌn+4โข(n-1)โ‰…โ„ฌnโˆ–โ„ฌnโข(n-5) for all large enough n. Eventually Dietrich [3] proved that indeed

โ„ฌnโข(n-4)โ‰…โ„ฌn+4โข(n-4)โ€ƒandโ€ƒโ„ฌn+4โข(n)โˆ–โ„ฌn+4โข(n-4)โ‰…โ„ฌnโข(n-4)โˆ–โ„ฌnโข(n-8)

for all large enough n; the proof that โ„ฌn+4โˆ–Bn+4โข(n)โ‰…โ„ฌnโˆ–โ„ฌnโข(n-4) for all large enough n is currently still missing. Nevertheless, these periodicity results describe ๐’ขโข(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 pโ‰ฅ7. Their analysis revealed that the structure of ๐’ขโข(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 ๐’ขโข(p,1) consisting of certain capable โ€œ1-parameter groupsโ€. Using the language of Section 4, these are the skeleton groups of the type Cn,eโข(๐•Šโข(๐’„)), where ๐’„=(c1,โ€ฆ,cโ„“) with exactly one non-zero ci. These subtrees of 1-parameter groups have finite widths if pโ‰ก5mod6, cf. the comment on [10, p. 299].

Newman and Oโ€™Brien [15] investigated the graph ๐’ขโข(2,r) for arbitrary r; their extensive computations led to the conjecture that each coclass tree in ๐’ขโข(2,r) is virtually periodic, that is, there is an integer dโ‰ฅ1 such that โ„ฌnโ‰…โ„ฌn+d for all large enough n.

The first periodicity theorem for general coclass graphs ๐’ขโข(p,r) was established independently by du Sautoy [18] and Eick and Leedham-Green [8]: they proved that for every coclass tree with branches โ„ฌ1,โ„ฌ2,โ€ฆ and every integer k>0, there exists dโ‰ฅ1 such that โ„ฌnโข(k)โ‰…โ„ฌn+dโข(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 ๐’ขโข(p,r) if and only if p=2 or (p,r)โˆˆ{(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 ๐’ขโข(p,1) in detail for pโ‰ก5mod6; 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 ๐’ขโข(5,1), which has finite width. For p>5 the coclass tree in ๐’ขโข(p,1) has infinite width, and the main result can be described as follows: First, there is an isomorphism of pruned branches โ„ฌnโข(n-2โขp+8)โ‰…โ„ฌn+p-1โข(n-2โขp+8) for all large enough n. Second, if G is a capable group at depth n-2โขp+8 in โ„ฌn+p-1 and if the automorphism group of its (p-1)-step parent H is a p-group, then ๐’Ÿp-1โข(G)โ‰…๐’Ÿp-1โข(H), where ๐’Ÿp-1โข(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 โ€œalmost allโ€ p-groups have a p-group as automorphism group, hence the results in [3] can be used to describe large parts of ๐’ขโข(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 โ„ฌn whose groups have automorphism group orders with constant pโ€ฒ-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 pโ‰ก5mod6 comes from underlying problems in p-adic number theory, already discussed in [10].

Eick, Leedham-Green, Newman and Oโ€™Brien [9] have investigated ๐’ขโข(3,2) in detail. More precisely, they have studied the skeletons of each of the sixteen coclass trees of ๐’ขโข(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 ๐’ขโข(p,r) from a finite subgroup. We briefly sketch this conjecture here; for full details see [9, Section 9]. Let ๐’ฏ be a coclass tree in ๐’ขโข(p,r) with branches โ„ฌ1,โ„ฌ2,โ€ฆ of arbitrarily large depths. Choose kโ‰ฅ1; the results in [8] imply that there exist dโ‰ฅ1 and lโˆˆโ„• such that Blโข(k)โ‰…Bl+iโขdโข(k) for all iโˆˆโ„•. For a group PโˆˆBl+iโขdโข(k) denote by Pยฏ the group in โ„ฌlโข(k) under a suitable graph isomorphism. Conjecture W now states that if one chooses k and l large enough, then there is a map ฮฝ from the groups of depth k in โ„ฌl to the groups of depth k-d in โ„ฌl such that the following holds: If P has depth k in โ„ฌl+iโขd, then ๐’Ÿโข(P)โ‰…๐’Ÿโข(Q), where Q is the group at depth k-d in โ„ฌl+(i-1)โขd corresponding to Qยฏ=ฮฝโข(Pยฏ), and ๐’Ÿโข(K) is the subtree generated by all descendants of K. This conjecture is illustrated in Figure 3.

Figure 3 An illustration of Conjecture W.

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.

References

[1] Blackburn N., On a special class of p-groups, Acta Math. 100 (1958), 45โ€“92. 10.1007/BF02559602Search in Google Scholar

[2] Couson M., Character degrees of finite p-groups by coclass, J. Algebra 418 (2014), 91โ€“109. 10.1016/j.jalgebra.2014.06.037Search in Google Scholar

[3] Dietrich H., A new periodic pattern in the graph of p-groups of maximal class, Bull. Lond. Math. Soc. 42 (2010), 1073โ€“1088. 10.1112/blms/bdq066Search in Google Scholar

[4] Dietrich H., Periodic patterns in the graph of p-groups of maximal class, J. Group Theory 13 (2010), 851โ€“871. 10.1515/jgt.2010.028Search in Google Scholar

[5] Dietrich H., Eick B. and Feichtenschlager D., Investigating p-groups by coclass with GAP, Computational Group Theory and the Theory of Groups (Davidson 2007), Contemp. Math. 470, American Mathematical Society, Providence (2008), 45โ€“61. 10.1090/conm/470/09185Search in Google Scholar

[6] Dixon J.โ€‰D., du Sautoy M.โ€‰P.โ€‰F., Mann A. and Segal D., Analytic Pro-p-groups, 2nd ed., Cambridge University Press, Cambridge, 2003. Search in Google Scholar

[7] Eick B., Metabelian p-groups and coclass theory, J. Algebra 421 (2015), 102โ€“118. 10.1016/j.jalgebra.2014.08.021Search in Google Scholar

[8] Eick B. and Leedham-Green C.โ€‰R., On the classification of prime-power groups by coclass, Bull. Lond. Math. Soc. 40 (2008), 274โ€“288. 10.1112/blms/bdn007Search in Google Scholar

[9] Eick B., Leedham-Green C.โ€‰R., Newman M.โ€‰F. and Oโ€™Brien E.โ€‰A., On the classification of groups of prime-power order by coclass: The 3-groups of coclass 2, Internat. J. Algebra Comput. 23 (2013), 1243โ€“1288. 10.1142/S0218196713500252Search in Google Scholar

[10] Leedham-Green C.โ€‰R. and McKay S., On the classification of p-groups of maximal class, Quart. J. Math. Oxford 35 (1984), 293โ€“304. 10.1093/qmath/35.3.293Search in Google Scholar

[11] Leedham-Green C.โ€‰R. and McKay S., The Structure of Groups of Prime Power Order, London Math. Soc. Monogr. Ser. (N.S.) 27, Oxford University Press, Oxford, 2002. Search in Google Scholar

[12] Leedham-Green C.โ€‰R. and Newman M. F., Space groups and groups of prime-power order I, Arch. Math. 35 (1980), 193โ€“203. 10.1007/BF01235338Search in Google Scholar

[13] Neukirch J., Algebraische Zahlentheorie, Springer, Berlin, 1992. 10.1007/978-3-540-37663-7Search in Google Scholar

[14] Newman M.โ€‰F., Groups of prime-power order, Groups (Canberra 1989), Lecture Notes in Math. 1456, Springer, Berlin (1990), 49โ€“62. 10.1007/BFb0100730Search in Google Scholar

[15] Newman M.โ€‰F. and Oโ€™Brien E.โ€‰A., Classifying 2-groups using coclass, Trans. Amer. Math. Soc. 351 (1990), 131โ€“169. 10.1090/S0002-9947-99-02124-8Search in Google Scholar

[16] Oโ€™Brien E.โ€‰A., The p-group generation algorithm, J. Symbolic Comput. 9 (1990), 677โ€“698. 10.1016/S0747-7171(08)80082-XSearch in Google Scholar

[17] Robinson D.โ€‰J.โ€‰S., Applications of cohomology groups to the theory of groups, Groups (St. Andrews 1981), London Math. Soc. Lecture Note Ser. 71, Cambridge University Press, Cambridge (1981), 46โ€“80. 10.1017/CBO9780511661884.005Search in Google Scholar

[18] du Sautoy M., Counting p-groups and nilpotent groups, Publ. Math. Inst. Hautes Etudes Sci. 92 (2001), 63โ€“112. 10.1007/BF02698914Search in Google Scholar

[19] Wiman A., รœber p-Gruppen mit maximaler Klasse, Acta Math. 88 (1952), 317โ€“346. 10.1007/BF02392136Search in Google Scholar

[20] GAP โ€“ Groups, Algorithms and Programming, http://www.gap-system.org. Search in Google Scholar

Received: 2016-3-15
Published Online: 2016-9-14
Published in Print: 2017-3-1

ยฉ 2017 by De Gruyter

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