Accessible 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’ข โˆ— 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’ข โˆ— 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’ข โˆ— for arbitrary p โ‰ฅ 7 . 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 graph ๐’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 G โ†’ H if and only if H / ฮณ โข ( H ) โ‰… G , where ฮณ โ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 p 2 , and an infinite coclass tree whose root is an elementary abelian group of order p 2 . This tree has a single infinite path starting at its root; the groups on this infinite path S 2 โ†’ S 3 โ†’ โ‹ฏ satisfy | S n | = p n 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 n โˆˆ โ„• the branch โ„ฌ n of ๐’ข is the subtree generated by all descendants of S n which are not descendants of S n + 1 . Note that every branch โ„ฌ n is a finite tree with root S n , 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„ฌ 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 โˆˆ โ„• the pruned branch โ„ฌ n โข ( k ) is the subtree of โ„ฌ n generated by all groups of depth at most k in โ„ฌ n . It was proved independently by du Sautoy [18] and Eick and Leedham-Green [8] that for each k โˆˆ โ„• there exists l โข ( k ) > 0 such that โ„ฌ n + p - 1 โข ( k ) โ‰… โ„ฌ n โข ( k ) for all n โ‰ฅ l โข ( k ) . Thus the pruned branches eventually repeat periodically; we call this the first periodicity.

If p โˆˆ { 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 = 2 โข p - 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’ข capable if it is not a leaf, and we define the skeleton ๐’ฎ n as the subtree of โ„ฌ n generated by all capable groups at depth at most n - m in โ„ฌ 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 ๐’ข has finite width if the number of groups of fixed depth in โ„ฌ n is bounded by a constant independent of n. It is known that ๐’ข has finite width if and only if p โ‰ค 5 . 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 p โ‰ฅ 7 . 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 n โ‰ฅ k the branch โ„ฌ n + p - 1 can be constructed from โ„ฌ 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 p โ‰ฅ 7 , which is fixed throughout this paper, and define constants

d = p - 1 , โ„“ = p - 3 2 , m = 2 โข p - 8 .

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

Theorem 1.1

Let n โ‰ฅ max โก { m , 8 } .

  1. (a)

    The skeleton ๐’ฎ n โˆ— has โ„“ groups at depth 1 ; we denote these by G n , 1 , โ€ฆ , G n , โ„“ .

  2. (b)

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

Theorem 1.1 exhibits that each skeleton ๐’ฎ n โˆ— 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 n โ‰ฅ max โก { m , 8 } and e < n - m . The number of groups at depth e in S n โˆ— is at least โ„“ โข p โŒŠ e / d โŒ‹ โข ( โ„“ - 1 ) ; in particular, G โˆ— has infinite width.

We have determined โ„ฌ n โˆ— for p = 7 and 10 โ‰ค n โ‰ค 18 by computer, cf. Section 1.2. Based on this, we formulate a conjectural description for โ„ฌ n โˆ— for arbitrary p โ‰ฅ 7 . To describe this conjecture, we define the twig ๐’ฒ G โˆ— of a group G in ๐’ฎ n โˆ— : this is the subtree of โ„ฌ n โˆ— with root G containing all descendants of G which are not descendants of any proper descendant of G in ๐’ฎ n โˆ— . Thus each group of โ„ฌ n โˆ— is contained in exactly one twig and the twigs of โ„ฌ n โˆ— are connected by the skeleton of โ„ฌ n โˆ— . We continue to denote the groups of depth 1 in a skeleton ๐’ฎ n โˆ— by G n , 1 , โ€ฆ , G n , 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’ฎ n โˆ— has exactly one of the groups G n , i as parent.

Conjecture 1.3

There exists l โ‰ฅ m such that for all n โ‰ฅ l and for each G of depth e at least 1 in ๐’ฎ n โˆ— with parent G n , i the following holds.

  1. (a)

    If G is not a leaf in ๐’ฎ n โˆ— , then the isomorphism type of the graph ๐’ฒ G โˆ— depends on the index i, on e mod d , and on n mod d only.

  2. (b)

    If G is a leaf in ๐’ฎ n โˆ— , then there exists a group G ยฏ with parent G n - d , i at depth e - d in ๐’ฎ n - d โˆ— with ๐’ฒ G โˆ— โ‰… ๐’ฒ G ยฏ โˆ— .

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

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„ฌ 10 โˆ— , โ€ฆ , โ„ฌ 16 โˆ— 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’ฎ n โˆ— .

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„ฌ n โˆ— (and skeleton ๐’ฎ n โˆ— ) 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’ฎ n , and we consider their isomorphism problem and automorphism groups. In Section 5 we then investigate the skeleton groups in ๐’ฎ n โˆ— 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„š p and โ„ค p denote the field of p-adic rational numbers and ring of p-adic integers, respectively. The p-th cyclotomic polynomial

1 + x + โ€ฆ + x p - 1 โˆˆ โ„š p โข [ x ]

is irreducible; we consider a fixed root ฮธ and define K = โ„š p โข ( ฮธ ) , so that K is a field extension of degree d = p - 1 over โ„š p , with โ„š p -basis { 1 , ฮธ , โ€ฆ , ฮธ d - 1 } . For a โˆˆ โ„ค with p โˆค a the field automorphism ฯƒ a : K โ†’ K is defined by ฯƒ a โข ( ฮธ ) = ฮธ a . The Galois group of K is Gal โข ( K ) = { ฯƒ a โˆฃ 1 โ‰ค a โ‰ค d } ; it is cyclic and we fix a generator ฯƒ = ฯƒ k . The equation order ๐’ช = โ„ค p โข [ ฮธ ] is the maximal order of K; it has { 1 , ฮธ , โ€ฆ , ฮธ d - 1 } as โ„ค p -basis, and a unique maximal ideal ๐”ญ = ( ฮธ - 1 ) . We abbreviate ฮบ = ฮธ - 1 , so that ( ฮบ m ) = ๐”ญ m for m โˆˆ โ„ค ; this defines a series of ideals through ๐’ช . For z โˆˆ โ„ค and n โˆˆ โ„• we denote by ( ๐”ญ z ) n the direct sum of n copies of ๐”ญ z (and not the ideal ๐”ญ n โข z ). For each non-zero w โˆˆ K there exist unique z โˆˆ โ„ค and a unit u โˆˆ โ„ค p โข [ ฮธ ] โˆ— such that w = ฮบ z โข u ; we call val โข ( w ) = z the valuation of w. Note that if v , w โˆˆ K are non-zero, then val โข ( v โข w ) = val โข ( v ) + val โข ( w ) . We extend this definition to non-zero n-tuples ๐’— โˆˆ K n , so that val โข ( ๐’— ) = z if and only if ๐’— โˆˆ ( ๐”ญ z ) n โˆ– ( ๐”ญ z + 1 ) n .

2.1 Eigenvalues

The generator ฯƒ of the Galois group of K can be considered as a โ„š p -linear map of K. The next lemma determines its eigenvalues; it is proved in [10, Lemma 2.3] for z โ‰ฅ 0 , but the same proof holds for all z โˆˆ โ„ค .

Lemma 2.1

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

2.2 The group of units

Let ๐’ฐ be the unit group of ๐’ช . For i โ‰ฅ 2 define ๐’ฐ i = 1 + ๐”ญ i , and let ฯ‰ be a primitive d-th root of unity in โ„ค p ; we assume throughout that ฯ‰ is chosen such that ฯ‰ โ‰ก k mod p where k is defined by the fixed generator ฯƒ = ฯƒ k of Gal โข ( K ) . It is shown in [13, Satz II.5.3] that

๐’ฐ = ใ€ˆ ฯ‰ ใ€‰ ร— ใ€ˆ ฮธ ใ€‰ ร— ๐’ฐ 2 .

The unit group of โ„ค p is โ„š p โˆฉ ๐’ฐ , that is, โ„ค p โˆ— = ใ€ˆ ฯ‰ ใ€‰ ร— ( 1 + p โข โ„ค p ) . 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‡ : U โ†’ U , u โ†ฆ u โข ฯƒ โข ( u ) - 1 , is a group homomorphism with ker โก ฯ‡ = Z p * and U = ker โก ฯ‡ ร— im โก ฯ‡ .

Proof.

Since ฯƒ generates Gal โข ( K ) , the fixed points of ฯƒ in K are exactly the elements of the subfield โ„š p of K, hence ker โก ฯ‡ = ๐’ฐ โˆฉ โ„š p = โ„ค p โˆ— . Next, we consider the restriction of ฯ‡ to ๐’ฐ 2 = 1 + ๐”ญ 2 . As shown in [13, Satz II.5.5 and p.โ€‰146], the additive group of ๐”ญ 2 is isomorphic to the multiplicative group ๐’ฐ 2 via

exp : ( ๐”ญ 2 , + ) โ†’ ( ๐’ฐ 2 , โ‹… ) , x โ†ฆ โˆ‘ k = 0 โˆž x k k ! .

As this exponential map is compatible with the action of ฯƒ, we can translate ฯ‡ to a map ฯˆ : ๐”ญ 2 โ†’ ๐”ญ 2 , x โ†ฆ x - ฯƒ โข ( x ) . Since ฯƒ is a diagonalisable โ„ค p -linear map on ๐”ญ 2 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 i โˆˆ { 1 , โ€ฆ , d - 1 } , then 1 - ฯ‰ i โ‰ข 0 mod p , so 1 - ฯ‰ i โˆˆ โ„ค p โˆ— ; moreover, Lemma 2.1 implies that there exists an eigenvector of ฯˆ in ๐”ญ 2 โˆ– ๐”ญ 3 . In conclusion, we have shown that

๐”ญ 2 = ker โก ฯˆ โŠ• im โก ฯˆ ,

and hence ๐’ฐ 2 is the direct product of ker โก ฯ‡ and im โก ฯ‡ restricted to ๐’ฐ 2 . Finally, note that ฯ‡ โข ( ฯ‰ ) = 1 and ฯ‡ โข ( ฮธ ) = ฮธ 1 - k . Thus the result follows from the decomposition of ๐’ฐ as ๐’ฐ = ใ€ˆ ฯ‰ ใ€‰ ร— ใ€ˆ ฮธ ใ€‰ ร— ๐’ฐ 2 . 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 = T โ‹Š P is an infinite pro-p-group of coclass 1.

The unique maximal S-invariant series through T is T = T 1 > T 2 > โ‹ฏ , where each T i = ( ๐”ญ i - 1 , + ) . Moreover, if i โ‰ฅ 2 , then T i = ฮณ i โข ( S ) is the i-th term in the lower central series of S. For i โˆˆ โ„• define S i = S / ฮณ i โข ( S ) , so that S 2 โ†’ S 3 โ†’ โ‹ฏ is the unique maximal infinite path in ๐’ข . The automorphism groups of S and its quotients S i are described in the following lemma from [4, Section 4.2]; it implies that the maximal path S 2 โ†’ S 3 โ†’ โ‹ฏ is contained in ๐’ข โˆ— .

Lemma 3.2

Writing elements of S as tuples ( t , ฮธ i ) with t โˆˆ T and i โˆˆ Z , the following hold:

  1. (a)

    The natural restriction ฯ€ : Aut โข ( S ) โ†’ Aut โข ( T ) , ฮฑ โ†ฆ ฮฑ | T , satisfies

    im โก ฯ€ โ‰… ๐’ฐ โ‹Š Gal โข ( K ) โ€ƒ ๐‘Ž๐‘›๐‘‘ โ€ƒ ker โก ฯ€ โ‰… Z 1 โข ( P , T ) .

    A preimage of ( u , ฯƒ b ) โˆˆ ๐’ฐ โ‹Š Gal โข ( K ) under ฯ€ is given by

    ฮฑ โข ( u , b ) : S โ†’ S , ( t , ฮธ i ) โ†ฆ ( u โข ฯƒ b โข ( t ) , ฮธ i โข b ) .

    The kernel of ฯ€ is generated by { ฮฑ โข ( 1 ) , ฮฑ โข ( ฮธ ) , โ€ฆ , ฮฑ โข ( ฮธ d - 1 ) } , where for s โˆˆ T we define

    ฮฑ โข ( s ) : S โ†’ S , ( t , ฮธ i ) โ†ฆ ( t + ( 1 + ฮธ + โ€ฆ + ฮธ i - 1 ) โข s , ฮธ i ) .

  2. (b)

    If i โ‰ฅ 4 , then the natural projection Aut โข ( S ) โ†’ Aut โข ( S i ) is surjective and | Out โข ( S i ) | = p i - 2 โข d 2 .

4 The skeleton groups in ๐’ฎ n

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 T โˆง T : this is the โ„ค p โข P -module generated by s โˆง t with s , t โˆˆ T such that for all s , s โ€ฒ , t โˆˆ T and z โˆˆ โ„ค p the following holds: s โˆง s = 0 , hence s โˆง t = - ( t โˆง s ) , and z โข ( s โˆง t ) = ( z โข s ) โˆง t = s โˆง ( z โข t ) , and ( s โˆง t ) + ( s โ€ฒ โˆง t ) = ( s + s โ€ฒ ) โˆง t . The group P acts diagonally on T โˆง T , which defines the โ„ค p โข P action on T โˆง T .

In the following let n โ‰ฅ max โก { m , 8 } and e โˆˆ { 0 , โ€ฆ , n - m } . Every surjective homomorphism f โˆˆ Hom P โข ( T โˆง T , T n ) defines an associative multiplication on T / T n + e via

( t + T n + e ) โˆ˜ ( s + T n + e ) = t + s + 1 2 โข f โข ( t โˆง s ) + T n + e .

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

C n , e โข ( f ) = M n , e โข ( f ) โ‹Š P ;

these groups are called constructible in [11]. Each group C n , e โข ( f ) is an extension of the natural S n -module T n / T n + e by the group S n on the infinite path of ๐’ข ; in particular, it is a group of depth e in the skeleton ๐’ฎ n . By [4, Theorem 1.3], the groups C n , e โข ( f ) are exactly the groups in the skeleton ๐’ฎ n .

4.1 The structure of Hom P โข ( T โˆง T , T )

The structure of Hom P โข ( T โˆง T , 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, T โˆง T = F โŠ• Z , where F is a โ„ค p โข P -module of rank โ„“ = ( p - 3 ) / 2 generated by ฮบ i โˆง ฮบ i - 1 with i = 1 , โ€ฆ , โ„“ , and Z is a free โ„ค p -module of rank 1. Let ฮด i โข j denote the Kronecker-delta and recall that ฯƒ a is an element of Gal โข ( K ) for p โˆค a . For i โˆˆ { 1 , โ€ฆ , โ„“ } we define ๐•‹ i โˆˆ Hom P โข ( T โˆง T , T ) via

๐•‹ i โข ( ฮบ j โˆง ฮบ j - 1 ) = ฮด i โข j โ€ƒ and โ€ƒ ๐•‹ i โข ( z ) = 0 โข for โข z โˆˆ Z .

For 2 โ‰ค a โ‰ค โ„“ + 1 we define ๐•Š a โˆˆ Hom P โข ( T โˆง T , T ) via

๐•Š a โข ( x โˆง y ) = ฯƒ a โข ( x ) โข ฯƒ 1 - a โข ( y ) - ฯƒ 1 - a โข ( x ) โข ฯƒ a โข ( y ) .

For an โ„“ -tuple ๐’„ = ( c 1 , โ€ฆ , c โ„“ ) โˆˆ K โ„“ let

๐•Š โข ( ๐’„ ) = c 1 โข ๐•Š 2 + โ€ฆ + c โ„“ โข ๐•Š โ„“ + 1 โ€ƒ and โ€ƒ ๐•‹ โข ( ๐’„ ) = c 1 โข ๐•‹ 1 + โ€ฆ + c โ„“ โข ๐•‹ โ„“ .

For a โˆˆ { 2 , โ€ฆ , โ„“ + 2 } and i โˆˆ { 1 , โ€ฆ , โ„“ } let

b a , i = ( ฮธ a - ฮธ 1 - a ) โข ( ( ฮธ a - 1 ) โข ( ฮธ 1 - a - 1 ) ) i - 1 ,

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

B = ( b 2 , 1 b 2 , 2 โ€ฆ b 2 , โ„“ b 3 , 1 b 3 , 2 โ€ฆ b 3 , โ„“ โ‹ฎ โ‹ฎ โ‹ฑ โ‹ฎ b โ„“ + 1 , 1 b โ„“ + 1 , 2 โ€ฆ b โ„“ + 1 , โ„“ ) โˆˆ K โ„“ 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 f โˆˆ Hom P โข ( T โˆง T , T ) .

  1. (a)

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

  2. (b)

    There exists a unique ๐’… โˆˆ K โ„“ with f = ๐•Š โข ( ๐’… ) .

  3. (c)

    The matrix B describes a base change from { ๐•Š 2 , โ€ฆ , ๐•Š โ„“ + 1 } to { ๐•‹ 1 , โ€ฆ , ๐•‹ โ„“ } , that is, ๐’„ = ๐’… โข B .

If ๐’„ โˆˆ ๐’ช โ„“ , then both ๐•‹ โข ( ๐’„ ) and ๐•Š โข ( ๐’„ ) lie in Hom P โข ( T โˆง T , T ) . By Lemma 4.1, { ๐•‹ 1 , โ€ฆ , ๐•‹ โ„“ } forms an ๐’ช -basis for Hom P โข ( T โˆง T , T ) . There exists ๐’„ โˆˆ K โ„“ โˆ– ๐’ช โ„“ with ๐•Š โข ( ๐’„ ) โˆˆ Hom P โข ( T โˆง T , T ) , which shows that the set { ๐•Š 2 , โ€ฆ , ๐•Š โ„“ + 1 } generates Hom P โข ( T โˆง T , 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’ฎ n can be obtained as C n , e โข ( ๐•Š โข ( ๐’„ ) ) , where ๐•Š โข ( ๐’„ ) has image T n ; thus we define

ฮ“ n = ( ๐”ญ n - 1 ) โ„“ โข B - 1 ,
ฮ” n = { ๐’„ โˆˆ K โ„“ โˆฃ C n , e โข ( ๐•Š โข ( ๐’„ ) ) โˆˆ ๐’ฎ n }
= ฮ“ n โˆ– ฮ“ n + 1 ,
ฮ” n , e * = { ๐’„ โˆˆ K โ„“ โˆฃ C n , e โข ( ๐•Š โข ( ๐’„ ) ) โˆˆ ๐’ฎ n โˆ— } โŠ† ฮ” n .

Note that ฮ” n = ฮบ โข ฮ” 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ญ ๐ข be the i-th row of B - 1 .

  1. (a)

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

  2. (b)

    If ๐’— = a 1 โข ๐’• ๐Ÿ + โ€ฆ + a โ„“ โข ๐’• โ„“ โˆˆ ฮ” 1 with each a i โˆˆ K , then

    val โข ( ๐’— ) = min โก { val โข ( a i โข ๐’• ๐’Š ) โˆฃ i = 1 , โ€ฆ , โ„“ } .

Proof.

(a) The proof of [11, Proposition 8.3.8] shows that det โก ( B ) โˆˆ ๐”ญ โ„“ 2 โˆ– ๐”ญ โ„“ 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•Š โข ( ๐’• ๐’Š ) ; it remains to analyse the valuation of ๐’• ๐’Š . It is straightforward to see that each entry b a , i of B has valuation val โข ( b a , i ) = ๐”ญ 2 โข i - 1 , hence

B = U โข diag โก ( ฮบ , ฮบ 3 , โ€ฆ , ฮบ 2 โข โ„“ - 1 ) ,

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

B - 1 = diag โก ( ฮบ - 1 , ฮบ - 3 , โ€ฆ , ฮบ - 2 โข โ„“ + 1 ) โข U - 1 .

Since det โก ( U ) โˆˆ ๐’ฐ , Cramerโ€™s rule for matrix inverses shows that each entry of U - 1 lies in ๐’ช . Since det โก ( U - 1 ) โˆˆ ๐’ฐ , each row and column of U - 1 contains at least one element in ๐’ช โˆ– 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 - 2 โข i + 1 .

(b) Let ฮผ = min โก { val โข ( a i โข ๐’• ๐’Š ) โˆฃ i = 1 , โ€ฆ , โ„“ } , J = { i โˆˆ 1 , โ€ฆ , โ„“ โˆฃ val โข ( a i โข ๐’• ๐’Š ) = ฮผ } . Clearly, val โข ( ๐’— ) โ‰ฅ ฮผ , and we have to show equality. If | J | = 1 , then val โข ( ๐’— ) = ฮผ follows readily; thus we suppose J = { i 1 , โ€ฆ , i n } with i 1 < โ€ฆ < i n and 1 < n . Note that val โข ( ๐’— ) = ฮผ if and only if

(4.1) val โข ( a i 1 โข ๐’• ๐’Š ๐Ÿ + โ€ฆ + a i n โข ๐’• ๐’Š ๐’ ) = ฮผ .

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

a i j โข ๐’• ๐’Š โ‰ก c i j โข ฮบ 2 โข i j - 1 โข ๐’• ๐’Š mod ( ๐”ญ ) โ„“ .

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

val โข ( a i 1 โข ๐’• ๐’Š ๐Ÿ + โ€ฆ + a i n โข ๐’• ๐’Š ๐’ ) > 0 .

This means that a i 1 โข ๐’• ๐’Š ๐Ÿ + โ€ฆ + a i n โข ๐’• ๐’Š ๐’ โˆˆ ( ๐”ญ ) โ„“ , and so

c i 1 โข ฮบ 2 โข i 1 - 1 โข ๐’• ๐’Š ๐Ÿ + โ€ฆ + c i n โข ฮบ 2 โข i n - 1 โข ๐’• ๐’Š ๐’ โˆˆ ( ๐”ญ ) โ„“ .

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

(4.2) c 1 โข ฮบ 1 โข ๐’• ๐Ÿ + c 2 โข ฮบ 3 โข ๐’• ๐Ÿ + โ€ฆ + c โ„“ โข ฮบ 2 โข โ„“ - 1 โข ๐’• 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 = diag โก ( ฮบ , ฮบ 3 , โ€ฆ , ฮบ 2 โข โ„“ - 1 ) โข B - 1 โˆˆ ๐’ช โ„“ 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 = B โข diag โก ( ฮบ - 1 , ฮบ - 3 , โ€ฆ , ฮบ - 2 โข โ„“ + 1 ) โˆˆ ๐’ช โ„“ ร— โ„“

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

B = diag โก ( x 2 , โ€ฆ , x โ„“ + 1 ) โข V โข ( z 2 , โ€ฆ , z โ„“ + 1 ) โข diag โก ( ฮบ 1 , ฮบ 3 , โ€ฆ , ฮบ 2 โข โ„“ - 1 ) ,

where

x a = ฮธ 1 - a โข ( 1 + ฮธ + โ€ฆ + ฮธ 2 โข a - 2 ) , z a = ( 1 + ฮธ + โ€ฆ + ฮธ a - 1 ) โข ( 1 + ฮธ + โ€ฆ + ฮธ - a ) โ€ƒ for all a ,

and V โข ( z 2 , โ€ฆ , z โ„“ + 1 ) is a Vandermonde matrix with parameters z 2 , โ€ฆ , z โ„“ + 1 . This yields

M - 1 = diag โก ( x 2 , โ€ฆ , x โ„“ + 1 ) โข V โข ( z 2 , โ€ฆ , z โ„“ + 1 ) .

Note that each x a โ‰ก ( 2 โข a - 1 mod p ) mod ๐”ญ โ‰  0 , and all z a โ‰ก ( - a 2 + a mod p ) mod ๐”ญ โ‰  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’ฎ n can be constructed as C = C n , e โข ( ๐•Š โข ( ๐’„ ) ) with ๐’„ โˆˆ ฮ” n . By definition, C is an extension of T n / T n + e by S n , and T n / T n + e is a fully invariant subgroup of C. Hence the isomorphism problem and the determination of Aut โ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’ฐ โ†’ ๐’ฐ , u โ†ฆ u - 1 โข ฯƒ a โข ( u ) โข ฯƒ 1 - a โข ( u ) ,

with a โˆˆ { 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 , ฯƒ b ) โˆˆ U โ‹Š Gal โข ( K ) acts on ๐œ = ( c 1 , โ€ฆ , c โ„“ ) โˆˆ ฮ” n via

( u , ฯƒ b ) โข ( ๐’„ ) = ( ฯ 2 โข ( u ) - 1 โข ฯƒ b โข ( c 1 ) , โ€ฆ , ฯ โ„“ + 1 โข ( u ) - 1 โข ฯƒ b โข ( c โ„“ ) ) โˆˆ ฮ” n .

This induces an action of U โ‹Š Gal โข ( K ) on ฮ“ n and on the set of cosets ฮ“ n / ฮ“ n + e for each e โˆˆ N .

Proof.

We explain the origin of this action. Every automorphism ฮฒ โˆˆ Aut โข ( S ) acts on f โˆˆ Hom P โข ( T โˆง T , T n ) via f โ†ฆ ฮฒ โข [ f ] = ฮฒ โˆ˜ f โˆ˜ ( ฮฒ - 1 โˆง ฮฒ - 1 ) | T โˆง T ; if f is surjective, then so is ฮฒ โข [ f ] . For the following, recall the notation of Lemma 3.2. If ฮฒ is in the kernel of the map ฯ€ : Aut โข ( S ) โ†’ Aut โข ( T ) , then ฮฒ โข [ f ] = f for every f โˆˆ Hom P โข ( T โˆง T , T n ) . Now let ฮฒ = ฮฑ โข ( u , b ) with ( u , ฯƒ b ) โˆˆ ๐’ฐ โ‹Š Gal โข ( K ) . If a โˆˆ { 2 , โ€ฆ , โ„“ + 1 } , c โˆˆ K , and x โˆง y โˆˆ T โˆง T , then a short computation shows that ฮฒ โข [ c โข ๐•Š a ] โข ( x โˆง y ) = ฯ a โข ( u - 1 ) โข ฯƒ b โข ( c ) โข ๐•Š a โข ( x โˆง y ) . This implies that f = ๐•Š โข ( ๐’„ ) is mapped to ฮฒ โข [ f ] = ๐•Š โข ( ( u , ฯƒ b ) โข ( ๐’„ ) ) , and ( u , ฯƒ b ) โข ( ๐’„ ) โˆˆ ฮ” 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 = C n , e โข ( ๐•Š โข ( ๐’„ ) ) be defined as above and let

ฮป : Aut โข ( S ) โ†’ Aut โข ( S n ) ร— Aut โข ( T n / T n + e ) ,
ฮป C : Aut โข ( C ) โ†’ Aut โข ( S n ) ร— Aut โข ( T n / T n + e )

be induced by the natural restrictions. It is easy to show that the kernel of ฮป C is isomorphic to Z 1 โข ( S n , T n / T n + e ) ; its image is described in [17] using cohomology. Here we describe the image of ฮป C in a different way that will be more useful in our setting. Recall from Lemma 3.2 that Aut โข ( S ) is an extension of Z 1 โข ( P , T ) by ๐’ฐ โ‹Š Gal โข ( K ) .

Theorem 4.4

Let n โ‰ฅ max โก { m , 8 } and e โˆˆ { 0 , โ€ฆ , n - m } ; let ๐œ , ๐ โˆˆ ฮ” n .

  1. (a)

    The groups C n , e โข ( ๐•Š โข ( ๐’„ ) ) and C n , e โข ( ๐•Š โ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’ฐ โ‹Š Gal โข ( K ) as defined in Lemma 4.3.

  2. (b)

    The group Aut โข ( C n , e โข ( ๐’„ ) ) is an extension of the p-group Z 1 โข ( S n , T n / T n + e ) by ฮป ( Z 1 ( P , T ) . ฮฃ ) , where ฮฃ is the stabiliser of ๐’„ + ฮ“ n + e in ๐’ฐ โ‹Š Gal โ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’ฐ โ‹Š Gal โข ( K ) .

(a) Every group of depth e in โ„ฌ n is an extension of T n / T n + e by the root S n of โ„ฌ n , where T n / T n + e carries the obvious S n -module structure, see [4, Theorem 3.1]. Such extensions can be described by elements of the second cohomology group H 2 โข ( S n , T n / T n + e ) , and the isomorphism problem of such extensions can be solved by considering the action of the group of compatible pairs Comp โข ( S n , T n / T n + e ) , which consists of pairs of compatible automorphisms of S n and T n / T n + 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 ฮป โข ( ฮฑ ) = ( ฮฑ | S n , ฮฑ | T n / T n + e ) defined by ฮฑ โˆˆ Aut โข ( S ) , acting on certain cohomology classes defined by surjective homomorphisms T โˆง T โ†’ T n . This action is discussed in detail in [3, Section 4.2], and it turns out that one has to consider exactly the automorphisms ฮฑ โข ( u , b ) of S which are defined by elements ( u , ฯƒ b ) โˆˆ ๐’ฐ โ‹Š Gal โข ( K ) ; the automorphisms ฮฑ โข ( s ) with s โˆˆ T act trivially. Putting all this together, the statement of the theorem follows. We note that in [4, 3] groups of depth e in โ„ฌ n are described as extensions of T / T e + 1 by S n ; however, T n / T n + e โ‰… T / T e + 1 as S n -modules, and applying a suitable S n -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 ฮณ โˆˆ H 2 โข ( S n , T n / T n + e ) be a cohomology class defining C n , e โข ( ๐•Š โข ( ๐’„ ) ) . It is shown in [17] that the image of ฮป C is isomorphic to the stabiliser of ฮณ in Comp โข ( S n , T n / T n + e ) , and that the kernel of ฮป C is isomorphic to the p-group Z 1 โข ( S n , T n / T n + 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 Aut โข ( S n + e ) , where ฮฑ โˆˆ Aut โข ( S n + e ) acts as the compatible pair ฮป โข ( ฮฑ ) . Note that Z 1 โข ( P , T ) . ฮฃ is the stabiliser of ๐’„ + ฮ“ n + e in Aut โข ( S ) ; since Aut โข ( S ) โ†’ Aut โข ( S n + 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’ฎ n โˆ—

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

Aut ( C ) โ‰… Z 1 ( S n , T n / T n + e ) . ฮป ( Z 1 ( P , T ) . ฮฃ ) ,

where ฮฃ is the stabiliser of ๐’„ + ฮ“ n + e in ๐’ฐ โ‹Š Gal โข ( K ) . Note that Z 1 โข ( S n , T n / T n + e ) and ฮป โข ( Z 1 โข ( P , T ) ) are both p-groups, and also the subquotient ฮป C โข ( ๐’ฐ 2 โˆฉ ฮฃ ) induces a p-group in Aut โข ( C ) . Recall that ๐’ฐ = ใ€ˆ ฯ‰ ใ€‰ ร— ใ€ˆ ฮธ ใ€‰ ร— ๐’ฐ 2 ; by Lemma 4.3, the element ฮธ โˆˆ ๐’ฐ acts trivially in Aut โข ( C ) , and ฯ‰ โˆˆ ๐’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 Aut โข ( C ) is an extension of a p-group by a subgroup of Gal โข ( K ) . Using the notation of Theorem 4.4, we have proved the following result; recall that Gal โข ( K ) is cyclic of order d = p - 1 and generated by ฯƒ.

Lemma 5.1

If n โ‰ฅ max โก { m , 8 } and e โˆˆ { 1 , โ€ฆ , n - m } , then

ฮ” n , e * = { ๐’„ โˆˆ ฮ” n โˆฃ there exists โข u โˆˆ ๐’ฐ โข with โข ( u , ฯƒ ) โข ( ๐’„ ) โ‰ก ๐’„ mod ฮ“ n + e } .

It remains to determine which units u โˆˆ ๐’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 y โˆˆ โ„ค ; recall that ฮ” n = ฮ“ n โˆ– ฮ“ n + 1 :

ฮฆ n , y = { ๐’„ โˆˆ ฮ“ n โˆฃ ( ฯ‰ y , ฯƒ ) โข ( ๐’„ ) = ๐’„ } ,
ฮ› n , y = { ๐’„ โˆˆ ฮ” n โˆฃ ( ฯ‰ y , ฯƒ ) โข ( ๐’„ ) = ๐’„ } .

It follows that ฮ› n , y = ฮฆ n , y โˆ– ฮฆ n + 1 , y for every y โˆˆ โ„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 e โˆˆ โ„• and y โˆˆ โ„ค we have

ฮ› n , y โŠ† ฮ” n , e * .

We can now state the main result of this section.

Theorem 5.2

Let ๐œ โˆˆ ฮ” n , e * where n โ‰ฅ max โก { m , 8 } and e โˆˆ { 1 , โ€ฆ , n - m } .

  1. (a)

    There exist y โˆˆ { ( n - 2 โข i ) mod d โˆฃ i = 1 , โ€ฆ , โ„“ } and ๐’‚ โˆˆ ฮ› n , y such that C n , e โข ( ๐•Š โข ( ๐’„ ) ) โ‰… C n , e โข ( ๐•Š โข ( ๐’‚ ) ) .

  2. (b)

    If ๐’„ โˆˆ ฮ› n , y 1 and ๐’… โˆˆ ฮ› n , y 2 with y 1 , y 2 โˆˆ โ„ค , then C n , e โข ( ๐•Š โข ( ๐’„ ) ) โ‰… C n , e โข ( ๐•Š โข ( ๐’… ) ) if and only if y 1 โ‰ก y 2 mod d and ๐’„ = ( u , 1 ) โข ( ๐’… ) mod ฮ“ n + e for some u โˆˆ โ„ค p * .

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 n โ‰ฅ max โก { m , 8 } and e โˆˆ { 1 , โ€ฆ , n - m } .

Lemma 5.3

If ๐œ , ๐ โˆˆ ฮ” n , e * , then

C n , e โข ( ๐•Š โข ( ๐’„ ) ) โ‰… C n , e โข ( ๐•Š โข ( ๐’… ) )

if and only if ( u , 1 ) โข ( ๐œ ) โ‰ก ๐ mod ฮ“ n + e for some u โˆˆ U .

Proof.

By Theorem 4.4, if C n , e โข ( ๐•Š โข ( ๐’„ ) ) โ‰… C n , e โข ( ๐•Š โข ( ๐’… ) ) , then

๐’… โ‰ก ( w , ฯƒ b ) โข ( ๐’„ ) mod ฮ“ n + e

for some ฯƒ b โˆˆ Gal โข ( K ) and w โˆˆ ๐’ฐ . By Lemma 5.1, there exists an element v โˆˆ ๐’ฐ with ( v , 1 ) โข ( ๐’„ ) โ‰ก ( 1 , ฯƒ ) โข ( ๐’„ ) mod ฮ“ n + e . Write ฯƒ b = ฯƒ h for some h โˆˆ { 1 , โ€ฆ , d } , and let u = w โข v h โˆˆ ๐’ฐ . Then

๐’… โ‰ก ( w , ฯƒ h ) โข ( ๐’„ ) โ‰ก ( w , 1 ) โข ( 1 , ฯƒ h ) โข ( ๐’„ ) โ‰ก ( w , 1 ) โข ( v h , 1 ) โข ( ๐’„ ) โ‰ก ( u , 1 ) โข ( ๐’„ ) mod ฮ“ n + e ,

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

Lemma 5.4

If ๐œ โˆˆ ฮ” n , e * , then C n , e โข ( S โข ( ๐œ ) ) โ‰… C n , e โข ( S โข ( ๐ ) ) for some ๐ โˆˆ ฮ” n , e * such that ( v , ฯƒ ) โข ( ๐ ) โ‰ก ๐ mod ฮ“ n + e for some v โˆˆ Z p * .

Proof.

By Lemma 5.1, there exists u โˆˆ ๐’ฐ with ( u , ฯƒ ) โข ( ๐’„ ) โ‰ก ๐’„ mod ฮ“ n + e . We use Lemma 2.2 to decompose u = v โข x with v โˆˆ ker โก ฯ‡ = โ„ค p * and x โˆˆ im โก ฯ‡ . Choose y โˆˆ ๐’ฐ with x - 1 = ฯ‡ โข ( y ) = y โข ฯƒ โข ( y ) - 1 , and define ๐’… = ( y , 1 ) โข ( ๐’„ ) . By Theorem 4.4, we have C n , e โข ( ๐’… ) โ‰… C n , e โ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 - 1 โข u , ฯƒ ) โข ( ๐’… )
โ‰ก ( v , ฯƒ ) โข ( ๐’… ) mod ฮ“ n + e . โˆŽ

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

(5.1) ฮฆ n , y = { ๐’„ โˆˆ ฮ“ n โˆฃ ( 1 , ฯƒ ) โข ( ๐’„ ) = ฯ‰ y โข ๐’„ } .

Lemma 5.5

For each y, the set ฮฆ n , y is a Z p -sublattice of ฮ“ n ; moreover

ฮ“ n = ฮฆ n , 0 โŠ• โ€ฆ โŠ• ฮฆ n , d - 1 .

Proof.

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

( x 1 , โ€ฆ , x โ„“ ) โ†ฆ ( ฯƒ โข ( x 1 ) , โ€ฆ , ฯƒ โข ( x โ„“ ) ) .

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

It follows from (5.1) that ฮฆ n , y = V y โˆฉ ฮ“ n , hence ฮฆ n , 0 โŠ• โ€ฆ โŠ• ฮฆ n , d - 1 โ‰ค ฮ“ n and each ฮฆ n , y is a sublattice of ฮ“ n . It remains to show the following inclusion: ฮ“ n โŠ† ฮฆ n , 0 โŠ• โ€ฆ โŠ• ฮฆ n , d - 1 . For this purpose, consider w โˆˆ ฮ“ n and write w = w 0 + โ€ฆ + w d - 1 with w i โˆˆ V i for i = 1 , โ€ฆ , โ„“ . Let k be the number of non-zero summands w i in w. We prove by induction on k that each w i โˆˆ ฮฆ n , 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 w j โ‰  0 . It follows from Lemma 4.3 that both ( 1 , ฯƒ ) โข ( w ) and ( ฯ‰ - j , 1 ) โข ( w ) = ฯ‰ j โข w lie in ฮ“ n , hence also their difference u = ( 1 , ฯƒ ) โข ( w ) - ฯ‰ j โข w lies in ฮ“ n . Note that u = u 0 + โ€ฆ + u d - 1 , where each u i = ( ฯ‰ i - ฯ‰ j ) โข w i โˆˆ V i . By construction, u has at most k - 1 non-zero summands u i and, by the induction hypothesis, we have u i โˆˆ ฮฆ n , i for each i. Recall that โ„ค p * = ใ€ˆ ฯ‰ ใ€‰ โข ( 1 + p โข โ„ค p ) ; this implies that ฯ‰ i - ฯ‰ j โ‰ข 0 mod p and hence ฯ‰ i - ฯ‰ j โˆˆ โ„ค p * for all i โ‰  j . Since u i = ( ฯ‰ i - ฯ‰ j ) โข w i โˆˆ ฮฆ n , i , it follows that w i โˆˆ ฮฆ n , i for all i โ‰  j . In particular, we have w - w j = w 0 + โ€ฆ + w j - 1 + w j + 1 + โ€ฆ + w d - 1 โˆˆ ฮ“ n , and so w j = w - ( w - w j ) โˆˆ ฮ“ n as well. Now clearly w j โˆˆ ฮฆ n , j , which completes the proof. โˆŽ

The next lemma considers ๐’… โˆˆ ฮ” n , e * and v โˆˆ โ„ค p * with ( v , ฯƒ ) โข ( ๐’… ) โ‰ก ๐’… mod ฮ“ n , e , as in Lemma 5.4. Recall that every v โˆˆ โ„ค p * can be written as

v = ฯ‰ u โข ( 1 + p โข x ) = ฯ‰ - y โข ( 1 + p โข x )

for some u , y โˆˆ { 0 , โ€ฆ , d - 1 } and x โˆˆ โ„ค p .

Lemma 5.6

Let ๐ โˆˆ ฮ” n , e * and v โˆˆ Z p * with ( v , ฯƒ ) โข ( ๐ ) โ‰ก ๐ mod ฮ“ n , e , and write v = ฯ‰ - y โข ( 1 + p โข x ) for some y โˆˆ { 0 , โ€ฆ , d - 1 } and x โˆˆ Z p . There exists ๐š โˆˆ ฮ› n , y with C n , e โข ( S โข ( ๐ ) ) = C n , e โข ( S โข ( ๐š ) ) .

Proof.

By definition, ๐’… โˆˆ ฮ” n = ฮ“ n โˆ– ฮ“ n + 1 ; by Lemma 5.5, we can decompose ๐’… = ๐’… 0 + โ€ฆ + ๐’… d - 1 with each ๐’… i โˆˆ ฮฆ n , i . As ( v , ฯƒ ) โข ( ๐’… ) โ‰ก ๐’… mod ฮ“ n + e and v โˆˆ โ„ค p โˆ— , it follows that

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

As v - 1 โข ฯ‰ i - 1 โˆˆ โ„ค p , Lemma 5.5 yields that ( v - 1 โข ฯ‰ i - 1 ) โข ๐’… i โˆˆ ฮ“ n + e for each i. Using v = ฯ‰ - y โข ( 1 + p โข x ) , it follows that v - 1 โข ฯ‰ i - 1 โ‰ก ฯ‰ i - y - 1 mod p ; in particular, v - 1 โข ฯ‰ i - 1 โˆˆ โ„ค p * and ๐’… i โˆˆ ฮ“ n + e for all i โ‰  y . We can now choose ๐’‚ = ๐’… y โˆˆ ฮฆ n , y ; we have shown that ๐’… โ‰ก ๐’‚ mod ฮ“ n + e , thus

C n , e โข ( ๐•Š โข ( ๐’… ) ) = C n , e โข ( ๐•Š โข ( ๐’‚ ) ) .

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

Lemma 5.7

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

Proof.

Let ๐’“ โˆˆ ฮ” 1 be an eigenvector of ( 1 , ฯƒ ) with eigenvalue ฯ‰ y - n + 1 . By Lemma 2.1, there exists an eigenvector x โˆˆ ๐”ญ n - 1 โˆ– ๐”ญ n of ฯƒ with eigenvalue ฯ‰ n - 1 . Then x โข ๐’“ โˆˆ ฮ” n and

( ฯ‰ y , ฯƒ ) โข ( x โข ๐’“ ) = ฯ‰ - y โข ฯƒ โข ( x ) โข ( 1 , ฯƒ ) โข ( ๐’“ )
= ฯ‰ - y โข ฯ‰ n - 1 โข x โข ฯ‰ y - n + 1 โข ๐’“
= x โข ๐’“ ,

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

( 1 , ฯƒ ) โข ( x โข ๐’„ ) = ฯ‰ - n + 1 โข x โข ( 1 , ฯƒ ) โข ( ๐’„ )
= ฯ‰ - n + 1 โข x โข ฯ‰ y โข ๐’„
= ฯ‰ y - n + 1 โข x โข ๐’„ . 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‰ - 2 โข i + 1 โˆฃ i = 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’‡ โˆˆ ฮ” 1 of ( 1 , ฯƒ ) is ฯ‰ val โข ( ๐’• ) , where val โข ( ๐’• ) is the valuation defined in Section 2. The eigenvalues on ฮ” 1 = ฮ“ 1 โˆ– 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 : C p โ„“ โ‰… ฮ“ 1 / ฮ“ 2 , ( a 1 , โ€ฆ , a โ„“ ) โ†ฆ a 1 โข ๐’• ๐Ÿ + โ€ฆ + a โ„“ โข ๐’• โ„“ + ฮ“ 2 ,

recall that each ๐’• ๐’‹ has valuation val โข ( ๐’• ๐’‹ ) = - 2 โข j + 1 . In the following we let ( 1 , ฯƒ ) act on C p โ„“ via ฯˆ; in particular, the action of ( 1 , ฯƒ ) on C p โ„“ is diagonalisable. Our claim is that for each i โˆˆ { 1 , โ€ฆ , โ„“ } there is, up to scalar multiples, a unique eigenvector ( a 1 , โ€ฆ , a i , 0 , โ€ฆ , 0 ) โˆˆ C p โ„“ of ( 1 , ฯƒ ) with a i โ‰  0 , and that this eigenvector has eigenvalue ฯ‰ - 2 โข i + 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 C p โ„“ is diagonalisable, there must be an eigenvector a = ( a 1 , โ€ฆ , a โ„“ ) โˆˆ C p โ„“ with a โ„“ โ‰  0 . This eigenvector comes from an eigenvector ๐’‡ = a 1 โข ๐’• ๐Ÿ + โ€ฆ + a โ„“ โข ๐’• โ„“ + ๐’ˆ โˆˆ ฮ” 1 for some ๐’ˆ โˆˆ ฮ” 2 . Recall that ฮ” 2 = ฮบ โข ฮ” 1 โŠ† ( ๐”ญ - 2 โข โ„“ + 2 ) โ„“ . Now

val โข ( a โ„“ โข ๐’• โ„“ ) = val โข ( a โ„“ ) + val โข ( ๐’• โ„“ ) = val โข ( ๐’• โ„“ ) = - 2 โข โ„“ + 1

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

b = a - a โ€ฒ = ( b 1 , โ€ฆ , b โ„“ - 1 , 0 ) โˆˆ C p โ„“

is an eigenvector of ( 1 , ฯƒ ) with eigenvalue ฯ‰ - 2 โข โ„“ + 1 . This eigenvector comes from an eigenvector ๐’– = b 1 โข ๐’• ๐Ÿ + โ€ฆ + b โ„“ - 1 โข ๐’• โ„“ - ๐Ÿ + ๐’‰ โˆˆ ฮ” 1 for some ๐’‰ โˆˆ ฮ” 2 . But now val โข ( ๐’– ) โ‰ฅ - 2 โข โ„“ + 2 , so the eigenvalue of b cannot be ฯ‰ - 2 โข โ„“ + 1 , a contradiction. (Indeed, if ฯ‰ val โข ( ๐’– ) = ฯ‰ - 2 โข โ„“ + 1 , then val โข ( ๐’– ) โ‰ฅ - 2 โข โ„“ + 2 forces val โข ( ๐’– ) โ‰ฅ - 2 โข โ„“ + p > 0 , so ๐’– โˆˆ ฮ” 2 and ( b 1 , โ€ฆ , b โ„“ - 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 C p โ„“ of the form ( a 1 , โ€ฆ , a โ„“ ) with a โ„“ โ‰  0 , and that the corresponding eigenvalue is ฯ‰ - 2 โข โ„“ + 1 .

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

v j = ( b j , 1 , โ€ฆ , b j , j , 0 , โ€ฆ , 0 ) โˆˆ C p โ„“

with b j , j โ‰  0 , and that the corresponding eigenvalue is ฯ‰ - 2 โข j + 1 . Note that v j comes from an eigenvector

๐’˜ ๐’‹ = b j , 1 โข ๐’• ๐Ÿ + โ€ฆ + b j , j โข ๐’• ๐’‹ + ๐’‰ ๐’‹ โˆˆ ฮ” 1

for some ๐’‰ ๐’‹ โˆˆ ฮ” 2 , with

val โข ( ๐’‰ ๐’‹ ) โ‰ฅ val โข ( b j , 1 โข ๐’• ๐Ÿ + โ€ฆ + b j , j โข ๐’• ๐’‹ ) = val โข ( b j , j โข ๐’• ๐’‹ ) = - 2 โข j + 1 .

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

a = ( a 1 , โ€ฆ , a i , 0 , โ€ฆ , 0 ) โˆˆ C p โ„“

of ( 1 , ฯƒ ) with a i โ‰  0 : if not, then there would be a basis of C p โ„“ consisting of { v i + 1 , โ€ฆ , v โ„“ } 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‰ - 2 โข i + 1 . Note that a comes from an eigenvector

๐’‡ = a 1 โข ๐’• ๐Ÿ + โ€ฆ + a i โข ๐’• ๐’Š + ๐’ˆ โˆˆ ฮ” 1

for some ๐’ˆ โˆˆ ฮ” 2 ; in the following, write r = val โข ( ๐’ˆ ) . If

r โ‰ฅ val โข ( a 1 โข ๐’• ๐Ÿ + โ€ฆ + a i โข ๐’• ๐’Š ) = val โข ( a i โข ๐’• ๐’Š ) = val โข ( a i ) + val โข ( a i โข ๐’• ๐’Š ) = val โข ( a i โข ๐’• ๐’Š ) = - 2 โข i + 1 ,

then val โข ( ๐’‡ ) = - 2 โข i + 1 by Lemma 4.2, and it follows that the eigenvalue is ฯ‰ - 2 โข i + 1 . It remains to consider the case r < - 2 โข i + 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 = u 1 โข ๐’• ๐Ÿ + โ€ฆ + u โ„“ โข ๐’• โ„“ โˆˆ ฮ” 2

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

๐’˜ ๐’” = b s , 1 โข ๐’• ๐Ÿ + โ€ฆ + b s , s โข ๐’• ๐’” + ๐’‰ ๐’” โˆˆ ฮ” 1

of ( 1 , ฯƒ ) with eigenvalue ฯ‰ - 2 โข s + 1 . Let k โˆˆ ๐”ญ r + 2 โข s - 1 โˆ– ๐”ญ r + 2 โข s be an eigenvector of ฯƒ with eigenvalue ฯ‰ r + 2 โข s - 1 . Now ๐’‡ โ€ฒ = ๐’‡ - k โข ๐’˜ ๐’” is an eigenvector of ( 1 , ฯƒ ) with eigenvalue ฯ‰ r , and that both ๐’‡ and ๐’‡ โ€ฒ correspond to

a = ( a 1 , โ€ฆ , a i , 0 , โ€ฆ , 0 ) โˆˆ C p โ„“

since k โˆˆ ๐”ญ . In particular,

๐’‡ โ€ฒ = a 1 โข ๐’• ๐Ÿ + โ€ฆ + a i โข ๐’• ๐’Š + ( ๐’ˆ - k โข ๐’˜ ๐’” )
= a 1 โข ๐’• ๐Ÿ + โ‹ฏ + a i โข ๐’• ๐’Š
+ ( ( u 1 - k โข b s , 1 ) โข ๐’• ๐Ÿ + โ‹ฏ + ( u s - k โข b s , s ) โข ๐’• ๐’” + u s + 1 โข ๐’• ๐’” + ๐Ÿ + โ‹ฏ + u โ„“ โข ๐’• โ„“ ) - k โข ๐’‰ ๐’” โŸ = โฃ : ๐’ˆ โ€ฒ .

Since both u s , k โข b s , s โˆˆ ๐”ญ r - 2 โข s + 1 โˆ– ๐”ญ r - 2 โข s + 2 , we can replace k by a suitable scalar multiple of k such that u s - k โข b s , s โˆˆ ๐”ญ r - 2 โข s + 2 , and so val โข ( ( u s - k โข b s , s ) โข ๐’• ๐’” ) > r ; note that val โข ( ( u j - k โข b s , j ) โข ๐’• ๐’‹ ) โ‰ฅ val โข ( u j โข ๐’• ๐’‹ ) for all j. In conclusion, we have found ๐’‡ โ€ฒ = a 1 โข ๐’• ๐Ÿ + โ€ฆ + a i โข ๐’• ๐’Š + ๐’ˆ โ€ฒ with ๐’ˆ โ€ฒ โˆˆ ฮ” 2 such that if ๐’ˆ โ€ฒ = u 1 โ€ฒ โข ๐’• ๐Ÿ + โ€ฆ + u โ„“ โ€ฒ โข ๐’• โ„“ and s โ€ฒ โˆˆ { 1 , โ€ฆ , โ„“ } is minimal with val โข ( u s โ€ฒ โข ๐’• ๐’” โ€ฒ ) = val โข ( ๐’ˆ โ€ฒ ) = val โข ( ๐’ˆ ) = r , then s โ€ฒ > s . (Note that ๐’‡ โ€ฒ has eigenvalue ฯ‰ r and r < - 2 โข i + 1 , so we must indeed have val โข ( ๐’ˆ โ€ฒ ) = r = val โข ( ๐’ˆ ) .) Now we iterate this argument until we find an eigenvector

๐’‡ ^ = a 1 โข ๐’• ๐Ÿ + โ€ฆ + a i โข ๐’• ๐’Š + ๐’ˆ ^

of ( 1 , ฯƒ ) with eigenvalue ฯ‰ r and ๐’ˆ ^ โˆˆ ฮ” 2 such that if ๐’ˆ ^ = u 1 ^ โข ๐’• ๐Ÿ + โ€ฆ + u โ„“ ^ โข ๐’• โ„“ , then val โข ( u j ^ โข ๐’• ๐’‹ ) > r for all j. But then Lemma 4.2 implies that val โข ( ๐’ˆ ^ ) > r , a contradiction to ฯ‰ val โข ( f ) = ฯ‰ r and val โข ( ๐’ˆ ^ ) = r . In summary, this proves r โ‰ฅ - 2 โข i + 1 , hence the eigenvalue of an eigenvector a = ( a 1 , โ€ฆ , a i , 0 , โ€ฆ , 0 ) with a i โ‰  0 must be ฯ‰ - 2 โข i + 1 .

Uniqueness. Consider a second eigenvector a โ€ฒ = ( a 1 โ€ฒ , โ€ฆ , a i โ€ฒ , 0 , โ€ฆ , 0 ) โˆˆ C p โ„“ of ( 1 , ฯƒ ) with a i โ€ฒ โ‰  0 ; as proved in the previous paragraph, the eigenvalue of a โ€ฒ is ฯ‰ - 2 โข i + 1 . We claim that a โ€ฒ and a are linearly dependent. Note that a comes from an eigenvector ๐’‡ = a 1 โข ๐’• ๐Ÿ + โ€ฆ + a i โข ๐’• ๐’Š + ๐’ˆ โˆˆ ฮ” 1 for some ๐’ˆ โˆˆ ฮ” 2 with val โข ( ๐’ˆ ) โ‰ฅ - 2 โข i + 1 . Similarly, a โ€ฒ comes from an eigenvector ๐’‡ โ€ฒ = a 1 โ€ฒ โข ๐’• ๐Ÿ + โ€ฆ + a i โ€ฒ โข ๐’• ๐’Š + ๐’ˆ โ€ฒ โˆˆ ฮ” 1 for some ๐’ˆ โ€ฒ โˆˆ ฮ” 2 with val โข ( ๐’ˆ โ€ฒ ) โ‰ฅ - 2 โข i + 1 . Suppose, for a contradiction, that a and a โ€ฒ are linearly independent. Replacing a โ€ฒ by a suitable scalar multiple, we can assume that a i = a i โ€ฒ , so that b = a - a โ€ฒ = ( b 1 , โ€ฆ , b j , 0 , โ€ฆ , 0 ) , with b j โ‰  0 and j < i , is also an eigenvector of ( 1 , ฯƒ ) with eigenvalue ฯ‰ - 2 โข i + 1 . This eigenvector comes from ๐’‡ ^ = ๐’‡ - ๐’‡ โ€ฒ = b 1 โข ๐’• ๐Ÿ + โ€ฆ + b j โข ๐’• ๐’‹ + ๐’ˆ ^ , where ๐’ˆ ^ = ๐’ˆ - ๐’ˆ โ€ฒ โˆˆ ฮ” 2 satisfies val โข ( ๐’ˆ ^ ) โ‰ฅ - 2 โข i + 1 ; in fact, we must have val โข ( ๐’ˆ ^ ) = - 2 โข i + 1 since otherwise the eigenvalue of b cannot be ฯ‰ - 2 โข i + 1 . Note that for all u = i + 1 , โ€ฆ , โ„“ we already found eigenvectors ๐’˜ ๐’– with eigenvalue ฯ‰ - 2 โข u + 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’‡ ~ = b 1 โข ๐’• ๐Ÿ + โ€ฆ + b j โข ๐’• ๐’‹ + ๐’ˆ ~ with eigenvalue ฯ‰ - 2 โข i + 1 , where ๐’ˆ ~ โˆˆ ฮ” 2 satisfies val โข ( ๐’ˆ ~ ) > - 2 โข i + 1 : this is not possible since ๐’‡ ~ has eigenvalue ฯ‰ - 2 โข i + 1 , that is, - 2 โข i + 1 = val โข ( ๐’‡ ~ ) = min โก { - 2 โข j + 1 , val โข ( ๐’ˆ ~ ) } , but we have deduced that min โก { - 2 โข j + 1 , val โข ( ๐’ˆ ~ ) } > - 2 โข i + 1 . This contradiction proves that a and a โ€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 ( a 1 , โ€ฆ , a i , 0 , โ€ฆ , 0 ) with a i โ‰  0 , and that the corresponding eigenvalue is ฯ‰ - 2 โข i + 1 . This completes the induction step. โˆŽ

Lemmas 5.7 and 5.8 yield the following corollary.

Corollary 5.9

We have ฮ› n , y โ‰  โˆ… if and only if y โ‰ก n - 2 โข i mod d for some i = 1 , โ€ฆ , โ„“ .

We can now prove Theorem 5.2โ€‰(a).

Proof of Theorem 5.2โ€‰(a).

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

C n , e โข ( ๐•Š โข ( ๐’„ ) ) โ‰… C n , e โข ( ๐•Š โข ( ๐’‚ ) ) .

Since ฮ› n , y โ‰  โˆ… , it follows from Corollary 5.9 that y โ‰ก n - 2 โข i mod d for some i โˆˆ { 1 , โ€ฆ , โ„“ } , as claimed. โˆŽ

5.2 Proof of Theorem 5.2โ€‰(b)

Again, we assume that n โ‰ฅ max โก { m , 8 } and e โˆˆ { 1 , โ€ฆ , n - m } . Recall the map ฯ‡ : ๐’ฐ โ†’ ๐’ฐ , u โ†ฆ u โข ฯƒ โข ( u ) - 1 , which is discussed in Lemma 2.2.

Lemma 5.10

Let ๐œ โˆˆ ฮ› n , y 1 and ๐ โˆˆ ฮ› n , y 2 with y 1 , y 2 โˆˆ Z . If C n , e โข ( S โข ( ๐œ ) ) โ‰… C n , e โข ( S โข ( ๐ ) ) , then y 1 = y 2 ; if u โˆˆ U with ๐ โ‰ก ( u , 1 ) โข ( ๐œ ) mod ฮ“ 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’„ ) mod ฮ“ n + e for some u โˆˆ ๐’ฐ . This implies that

๐’… โ‰ก ( ฯ‰ y 2 , ฯƒ ) โข ( ๐’… )
โ‰ก ( ฯ‰ y 2 , ฯƒ ) โข ( u , 1 ) โข ( ๐’„ )
โ‰ก ( ฯ‰ y 2 , ฯƒ ) โข ( u , 1 ) โข ( ฯ‰ y 1 , ฯƒ ) - 1 โข ( ๐’„ )
โ‰ก ( ฯ‰ y 2 , ฯƒ ) โข ( u , 1 ) โข ( ฯ‰ y 1 , ฯƒ ) - 1 โข ( u , 1 ) - 1 โข ( ๐’… )
โ‰ก ( ฯ‰ y 2 - y 1 โข ฯƒ โข ( u ) โข u - 1 , 1 ) โข ( ๐’… )
โ‰ก ( ฯ‰ y 2 - y 1 โข ฯ‡ โข ( u ) - 1 , 1 ) โข ( ๐’… ) mod ฮ“ n + e .

By Theorem 4.4, the element ( ฯ‰ y 2 - y 1 โข ฯ‡ โข ( u ) - 1 , 1 ) yields an element of the group Aut โข ( C n , e โข ( ๐’… ) ) , and the discussion of the automorphism groups of skeleton groups (in the beginning of Section 5) forces that y 2 โ‰ก y 1 mod d . Since we have y 1 , y 2 โˆˆ { 0 , โ€ฆ , d - 1 } , this yields y 1 = y 2 . 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 โˆˆ ฮ› n , y with y โˆˆ Z . If C n , e โข ( S โข ( ๐œ ) ) โ‰… C n , e โข ( S โข ( ๐ ) ) , then ๐œ โ‰ก ( s , 1 ) โข ( ๐ ) mod ฮ“ n + e for some s โˆˆ Z p * .

Proof.

Recall that ๐’„ , ๐’… โˆˆ ฮ› n , y โŠ† ฮ” n , e * . Thus, by Lemma 5.3, there exists u โˆˆ ๐’ฐ with ( u , 1 ) โข ( ๐’„ ) โ‰ก ๐’… mod ฮ“ 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‡ | im โก ฯ‡ induces an automorphism of im โก ฯ‡ ; write u = s โข w with s โˆˆ ker โก ฯ‡ = โ„ค p * and w โˆˆ im โก ฯ‡ . We want to show that ( w , 1 ) โข ( ๐’… ) โ‰ก ๐’… mod ฮ“ 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’… ) mod ฮ“ n + e ,

which proves the lemma.

We start with a more general observation: Let v โˆˆ ๐’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 ( ฯƒ โข ( v ) , 1 ) โข ( ฯ‰ y , ฯƒ ) โข ( ๐’… ) โ‰ก ( ฯƒ โข ( v ) , 1 ) โข ( ๐’… ) mod ฮ“ 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‡ 2 โข ( w ) , 1 ) stabilises ๐’… modulo ฮ“ n + e . By induction, ( ฯ‡ i โข ( w ) , 1 ) stabilises ๐’… modulo ฮ“ n + e for every i โ‰ฅ 1 .

There is j โ‰ฅ 2 such that ๐’ฐ j = 1 + ๐”ญ j acts trivially on ฮ“ n / ฮ“ n + e : for example, choose j large enough such that ๐”ญ j โ‰ค p x โข ๐’ช for some x with p x โข ฮ“ n โ‰ค ฮ“ n + e . It follows from the definition that ฯ‡ stabilises ๐’ฐ j . Since ๐’ฐ j โ‰ค ๐’ฐ 2 , the proof of Lemma 2.2 shows that ๐’ฐ j = ker โก ( ฯ‡ | ๐’ฐ j ) ร— im โก ( ฯ‡ | ๐’ฐ j ) ; this implies that ฯ‡ induces an automorphism of J = ( im โก ฯ‡ ) / ( im โก ฯ‡ | ๐’ฐ j ) , 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 im โก ฯ‡ . If w lies in ๐’ฐ j , then ( w , 1 ) acts trivially on ๐’… modulo ฮ“ n + e , and there is nothing to show. If w โˆ‰ ๐’ฐ j , then its coset c = w โข im โก ( ฯ‡ | ๐’ฐ j ) in J is non-trivial, and ฯˆ t โข ( c ) = c follows. But this means that ฯ‡ t โข ( w ) = w โข r for some r โˆˆ ๐’ฐ j . As shown above, ฯ‡ t โข ( w ) = w โข r stabilises ๐’… modulo ฮ“ n + e . Since r โˆˆ ๐’ฐ j 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’„ โˆˆ ฮ› n , y 1 and ๐’… โˆˆ ฮ› n , y 2 . If there exists an element u โˆˆ โ„ค p * with ๐’„ โ‰ก ( u , 1 ) โข ( ๐’… ) mod ฮ“ n + e , then we have C n , e โข ( ๐•Š โข ( ๐’„ ) ) โ‰… C n , e โข ( ๐•Š โข ( ๐’… ) ) by Lemma 5.3. For the converse, suppose that C n , e โข ( ๐•Š โข ( ๐’„ ) ) โ‰… C n , e โข ( ๐•Š โข ( ๐’… ) ) . Then y 1 โ‰ก y 2 mod d follows from Lemma 5.10, and ๐’„ = ( u , 1 ) โข ( ๐’… ) mod ฮ“ n + e for some