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18 results

  • Structure of blocks with normal defect and abelian $p'$ inertial quotient

  • Part of
  • David Benson, Radha Kessar, Markus Linckelmann
  • Journal:
    Forum of Mathematics, Sigma / Volume 11 / 2023
    Published online by Cambridge University Press:
    01 March 2023, e13
    • Article
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  • Let k be an algebraically closed field of prime characteristic p. Let $kGe$ be a block of a group algebra of a finite group G, with normal defect group P and abelian $p'$ inertial quotient L. Then we show that $kGe$ is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings–Quillen style theorem.

    As an example, we calculate the associated graded of the basic algebra of the nonprincipal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order $p^3$ with a quaternion group of order eight with the centre acting trivially. In the case of $p=3$ , we give explicit generators and relations for the basic algebra as a quantised version of $kP$ . As a second example, we give explicit generators and relations in the case of a group of shape $2^{1+4}:3^{1+2}$ in characteristic two.

Fusion Systems in Algebra and Topology
  • Michael Aschbacher, Radha Kessar, Bob Oliver
  • Published online:
    07 September 2011
    Print publication:
    25 August 2011
  • A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.

  • Part II - The local theory of fusion systems

  • Michael Aschbacher, California Institute of Technology, Radha Kessar, University of Aberdeen, Bob Oliver, Université de Paris XIII
  • Book:
    Fusion Systems in Algebra and Topology
    Published online:
    07 September 2011
    Print publication:
    25 August 2011, pp 49-102

  • Summary

    Part II of the book is intended to be an introduction to the local theory of saturated fusion systems. By the “local theory of fusion systems” we mean an extension of some part of the local theory of finite groups to the setting of saturated fusion systems on finite p-groups.

    Recall the notions of a p-local finite group and its associated linking system defined in Definitions III.4.1 and III.4.4. One can ask: Why deal with saturated fusion systems rather than p-local finite groups? There are two reasons for this choice. First, it is not known whether to each saturated fusion system there is associated a unique p-local finite group. Thus it remains possible that the class of saturated fusion systems is larger than the class of p-local finite groups. But more important, to date there is no accepted notion of a morphism of p-local finite groups, and hence no category of p-local groups. (The problem arises already for fusion systems and linking systems of groups, since a group homomorphism α from G to H need not send p-centric subgroups of G to p-centric subgroups of H, so it is not clear how to associate to α a map which would serve as a morphism of the p-local group of G with the p-local group of H.) The local theory of finite groups is inextricably tied to the notion of group homomorphism and factor group, so to extend the local theory of finite groups to a different category, we must at the least be dealing with an actual category.

  • Appendix A - Background facts about groups

  • Michael Aschbacher, California Institute of Technology, Radha Kessar, University of Aberdeen, Bob Oliver, Université de Paris XIII
  • Book:
    Fusion Systems in Algebra and Topology
    Published online:
    07 September 2011
    Print publication:
    25 August 2011, pp 300-305

  • Summary

    We collect here, for reference, some of the standard results in group theory which have been used throughout the book. We begin with some of the standard elementary properties of p-groups and Sylow subgroups.

    Lemma A.1For any pair of p-groups P ≤ Q, if NQ(P) = P, then Q = P.

    Proof. This is a general property of nilpotent groups (finite or not). If G is nilpotent and H < G, then by definition, there is K > H such that [G, K] ≤ H. Hence H < K ≤ NG(H). That all p-groups are nilpotent is shown, e.g., in [A4, 9.8] or [G1, Theorem 2.3.3].

    Lemma A.2Fix a p-group P, and an automorphism α ∈ Aut(P) of order prime to p. Assume 1 = P0 ⊴ P1 ⊴ … ⊴ Pm = P is a sequence of subgroups all normal in P, such that for each 1 ≤ i ≤ m,α|Pi ≡ IdPi (mod Pi-1). Thenα = IdP.

    Proof. See, for example, [G1, Theorem 5.3.2]. It suffices by induction to prove this when m = 2, and when the order of α is a prime q ≠ p. In this case, for each g ∈ P, α acts on the coset gP1 with fixed subset of order ≡ |gP1| (mod q). Since |gP1| = |P1| is a power of p, this shows that α fixes at least one element in gP1. Thus α is the identity on P1 and on at least one element in each coset of P1, and so α = IdP.

  • Introduction

  • Michael Aschbacher, California Institute of Technology, Radha Kessar, University of Aberdeen, Bob Oliver, Université de Paris XIII
  • Book:
    Fusion Systems in Algebra and Topology
    Published online:
    07 September 2011
    Print publication:
    25 August 2011, pp 1-4

  • Summary

    Let G be a finite group, p a prime, and S a Sylow p-subgroup of G. Subsets of S are said to be fused in G if they are conjugate under some element of G. The term “fusion” seems to have been introduced by Brauer in the fifties, but the general notion has been of interest for over a century. For example, in his text The Theory of Groups of Finite Order [Bu] (first published in 1897), Burnside proved that if S is abelian then the normalizer in G of S controls fusion in S. (A subgroup H of G is said to control fusion in S if any pair of tuples of elements of S which are conjugate in G are also conjugate under H.)

    Initially, information about fusion was usually used in conjunction with transfer, as in the proof of the normal p-complement theorems of Burnside and Frobenius, which showed that, under suitable hypotheses on fusion, G possesses a normal p-complement: a normal subgroup of index |S| in G. But in the sixties and seventies more sophisticated results on fusion began to appear, such as Alperin's Fusion Theorem [Al1], which showed that the family of normalizers of suitable subgroups of S control fusion in S, and Goldschmidt's Fusion Theorem [Gd3], which determined the groups G possessing a nontrivial abelian subgroup A of S such that no element of A is fused into S\A.

    In the early nineties, Lluis Puig abstracted the properties of G-fusion in a Sylow subgroup S, in his notion of a Frobenius category on a finite p-group S, by discarding the group G and focusing instead on isomorphisms between subgroups of S.

  • BLOCKS WITH A QUATERNION DEFECT GROUP OVER A 2-ADIC RING: THE CASE Ã4

  • THORSTEN HOLM, RADHA KESSAR, MARKUS LINCKELMANN
  • Journal:
    Glasgow Mathematical Journal / Volume 49 / Issue 1 / January 2007
    Published online by Cambridge University Press:
    01 January 2007, pp. 29-43
    Print publication:
    January 2007
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  • Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field of prime characteristic determines that of the corresponding block algebra over a p-adic ring. We prove this to be the case when the defect group is quaternion of order 8 and the block algebra over an algebraically closed field k of characteristic 2 is Morita equivalent to kÃ4. The main ingredients are Erdmann's classification of tame blocks [6] and work of Cabanes and Picaronny [4, 5] on perfect isometries between tame blocks.