Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- An army of cohomology against residual finiteness
- On some questions concerning subnormally monomial groups
- A conjecture concerning the evaluation of products of class-sums of the symmetric group
- Automorphisms of Burnside rings
- On finite generation of unit groups for group rings
- Counting finite index subgroups
- The quantum double of a finite group and its role in conformal field theory
- Closure properties of supersoluble Fitting classes
- Groups acting on locally finite graphs - a survey of the infinitely ended case
- An invitation to computational group theory
- On subgroups, transversals and commutators
- Intervals in subgroup lattices of finite groups
- Amalgams of minimal local subgroups and sporadic simple groups
- Vanishing orbit sums in group algebras of p-groups
- From stable equivalences to Rickard equivalences for blocks with cyclic defect
- Factorizations in which the factors have relatively prime orders
- Some problems and results in the theory of pro-p groups
- On equations in finite groups and invariants of subgroups
- Group presentations where the relators are proper powers
- A condensing theorem
- Lie methods in group theory
- Some new results on arithmetical problems in the theory of finite groups
- Groups that admit partial power automorphisms
- Problems
Automorphisms of Burnside rings
Published online by Cambridge University Press: 19 February 2010
- Frontmatter
- Contents
- Preface
- Introduction
- An army of cohomology against residual finiteness
- On some questions concerning subnormally monomial groups
- A conjecture concerning the evaluation of products of class-sums of the symmetric group
- Automorphisms of Burnside rings
- On finite generation of unit groups for group rings
- Counting finite index subgroups
- The quantum double of a finite group and its role in conformal field theory
- Closure properties of supersoluble Fitting classes
- Groups acting on locally finite graphs - a survey of the infinitely ended case
- An invitation to computational group theory
- On subgroups, transversals and commutators
- Intervals in subgroup lattices of finite groups
- Amalgams of minimal local subgroups and sporadic simple groups
- Vanishing orbit sums in group algebras of p-groups
- From stable equivalences to Rickard equivalences for blocks with cyclic defect
- Factorizations in which the factors have relatively prime orders
- Some problems and results in the theory of pro-p groups
- On equations in finite groups and invariants of subgroups
- Group presentations where the relators are proper powers
- A condensing theorem
- Lie methods in group theory
- Some new results on arithmetical problems in the theory of finite groups
- Groups that admit partial power automorphisms
- Problems
Summary
The object of this article is the Burnside ring Ω(G) for a finite group G. If G is soluble we show how the group of ring automorphisms Aut(Ω(G)) may be calculated purely from the knowledge of the subgroup lattice of G. The results on Aut(Ω(G)) for an abelian group G by Krämer [6] and results of Lezaun [7] for dihedral and some special metacyclic groups follow as special cases. For general facts on Burnside rings we refer to [2, §80].
The paper is organized as follows: In Section 1 we recall some more or less well known facts about the table of marks, the ghost ring of Ω(G) and Aut(Ω(G)). We introduce the group of normalized automorphism Autn(Ω(G)) as the automorphisms which stabilize the regular G – set G/l. In (2.4) we reduce the study of Aut(Ω(G)) for soluble groups to the study of Autn(Ω(G)), extending a result of Krämer [6] from nilpotent groups to soluble groups. Any automorphism a of Ω(G) induces in a natural way – via the ghost ring – a bijection σ* on the set V(G) of conjugacy classes of subgroups of G. In (3.1) it is shown that this is indeed an automorphism of V(G) considered in a special way as a partially ordered set, provided σ is augmented and G is soluble. In (3.2) we characterize those σ ∈ Aut(Ω(G)), which send transitive G-sets to transitive G-sets, in terms of the map σ*. In (3.6) we give an explicit constructive description of Autn(Ω(G)) in terms of the automorphisms of V(G) as a partially ordered set; i.e. for a fixed group it allows one to compute explicitly Autn(Ω(G)) with the help of a computer algebra system.
- Type
- Chapter
- Information
- Groups '93 Galway/St Andrews , pp. 333 - 351Publisher: Cambridge University PressPrint publication year: 1995