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Page 1 of 3

  • SOME INFINITE PERMUTATION GROUPS AND RELATED FINITE LINEAR GROUPS

  • Part of
  • PETER M. NEUMANN, CHERYL E. PRAEGER, SIMON M. SMITH
  • Journal:
    Journal of the Australian Mathematical Society / Volume 102 / Issue 1 / February 2017
    Published online by Cambridge University Press:
    25 October 2016, pp. 136-149
    Print publication:
    February 2017
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  • This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$ -group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$ -adic vector space associated with  $M$ . This leads to our second variation, which is a study of the finite linear groups that can arise.

  • A note on the triple product property for subsets of finite groups

  • Part of
  • Peter M. Neumann
  • Journal:
    LMS Journal of Computation and Mathematics / Volume 14 / 2011
    Published online by Cambridge University Press:
    01 September 2011, pp. 232-237
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  • The triple product property (TPP) for subsets of a finite group was introduced by Henry Cohn and Christopher Umans in 2003 as a tool for the study of the complexity of matrix multiplication. This note records some consequences of the simple observation that if (S1,S2,S3) is a TPP triple in a finite group G, then so is (dS1a,dS2b,dS3c) for any a,b,c,d∈G.

    Let si:=∣Si∣ for 1≤i≤3. First we prove the inequality s1(s2+s3−1)≤∣G∣ and show some of its uses. Then we show (something a little more general than) that if G has an abelian subgroup of index v, then s1s2s3 ≤v2 ∣G∣.

  • On Tensor-Factorisation Problems,I: The Combinatorial Problem

  • Peter M. Neumann, Cheryl E. Praeger
  • Journal:
    LMS Journal of Computation and Mathematics / Volume 7 / 2004
    Published online by Cambridge University Press:
    01 February 2010, pp. 73-100
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  • A k-multiset is an unordered k-tuple, perhaps with repetitions. If x is an r-multiset {x1, …, xr} and y is an s-multiset {y1, …, ys} with elements from an abelian group A the tensor product x ⊗ y is defined as the rs-multiset {xi yj | 1 ≤ i ≤ r, 1 ≤ j ≤ s}. The main focus of this paper is a polynomial-time algorithm to discover whether a given rs-multiset from A can be factorised. The algorithm is not guaranteed to succeed, but there is an acceptably small upper bound for the probability of failure. The paper also contains a description of the context of this factorisation problem, and the beginnings of an attack on the following division-problem: is a given rs-multiset divisible by a given r-multiset, and if so, how can division be achieved in polynomially bounded time?

Enumeration of Finite Groups
  • Simon R. Blackburn, Peter M. Neumann, Geetha Venkataraman
  • Published online:
    12 January 2010
    Print publication:
    18 October 2007
  • How many groups of order n are there? This is a natural question for anyone studying group theory, and this Tract provides an exhaustive and up-to-date account of research into this question spanning almost fifty years. The authors presuppose an undergraduate knowledge of group theory, up to and including Sylow's Theorems, a little knowledge of how a group may be presented by generators and relations, a very little representation theory from the perspective of module theory, and a very little cohomology theory - but most of the basics are expounded here and the book is more or less self-contained. Although it is principally devoted to a connected exposition of an agreeable theory, the book does also contain some material that has not hitherto been published. It is designed to be used as a graduate text but also as a handbook for established research workers in group theory.

  • 16 - Pyber's theorem: the general case

  • from III - PYBER'S THEOREM
  • Simon R. Blackburn, Royal Holloway, University of London, Peter M. Neumann, University of Oxford, Geetha Venkataraman, University of Delhi
  • Book:
    Enumeration of Finite Groups
    Published online:
    12 January 2010
    Print publication:
    18 October 2007, pp 140-160

  • Summary

    In this chapter we aim to prove the general version of Pyber's theorem: the proof is contained in the final section. The three sections preceding the proof each deal with a different ingredient that is needed there. Section 16.1 contains theorems that bound the number of generators of a group in various contexts. Section 16.2 is concerned with central extensions (especially of perfect groups). Finally, in Section 16.3 we define and explore the notion of the generalised Fitting subgroup of a group.

    Three theorems on group generation

    This section contains proofs of three theorems, each of which makes statements about the existence of certain kinds of generating sets for finite groups. The first, due to Wolfgang Gaschütz, [35], will be needed to prove the third theorem of this section. The second and third depend on the Classification of Finite Simple Groups; they will be used in the proof of the general case of Pyber's theorem in Section 16.4.

    Theorem 16.1Let G be a finite group, and let N be a normal subgroup of G. Suppose that G may be generated by r elements, and let g1, g2 …, gr ∈ G be such that g1N, g2N, …, grN generate G/N. Then there exist generators {h1, h2 …, hr} for G such that hi ∈ giN for i ∈ {1, 2, … r}.

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