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
Problems
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
Problem 1. Let F be the free group of finite rank n ≥ 2. Suppose an endomorphism φ of F takes every primitive element to a primitive one. Is it true that φ is actually an automorphism of F?
For n = 2, the answer was proved to be “yes” by S. Ivanov.
Vladimir Shpilrain
Problem 2. (Closure operators and finite solvable groups) Let G be a finite group, let L(G) be its lattice of subgroups and let σ : L → L(G) be a closure operator on L(G). This means that:
(i) if H, K ≤ G, then σ(H) ≤ σ(K);
(ii) if H ≤ G, then H ≤ σ(H); and
(iii) if H ≤ G, then σ(H) = σ(σ(H)).
Assume now that σ is a non–trivial closure operator, i.e. σ can be defined for every L(G), where G is a finite group and there exists a finite group X such that σ : L(X) → (X) is not trivial. The following problem arises: if σ : L(G) → L(G) is as above and if every subgroup of G is closed, i.e. σ(H) = H, ∀H ≤ G, is it true that G is solvable?
Arguments for the “yes” answer:
If σ(H) := HG = 〈Hg|g ∈ G〉 and if σ(H) = H, ∀H ≤ G, then G is Dedekind, so nilpotent.
If σ(H) := CG(CG(H)) and if σ(H) = H, ∀H, ≤ G, then G is supersolvable (Gaschütz).
If σ(H) := ∩{M|M is a maximal subgroup of G and H ≤ M} and if σ(H) = H, ∀H ≤ G, then G is supersolvable (Menegazzo, di Martino et al.)
- Type
- Chapter
- Information
- Groups '93 Galway/St Andrews , pp. 602 - 609Publisher: Cambridge University PressPrint publication year: 1995