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
Factorizations in which the factors have relatively prime orders
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
We shall assume throughout that G is a finite abelian group. We say that G is the direct sum of subsets A, B if each g ∈ G can be expressed uniquely as g = a + b, a ∈ A, b ∈ B. We write G = A + B and call this a factorization of G. Clearly |G| = |A||B|, where |X| denotes the order of a subset X of G.
If |A|, |B| are relatively prime then G possesses unique subgroups H, K with |A| = |H| and |B| = |K|. G is the direct sum of these subgroups. In [3] the question was raised whether, under these conditions, G = A + B implies either that G = H + B or that G = A + K. It is shown there that if A has prime power order then G = A + K. In [4] it is shown that if G is cyclic and A has prime power order then also G = H + B. It is also shown in [4] that if A has order 6 then G = H + B.
In this note we generalise these last two results. In the first case we show that the result holds whenever the p-component of G is cyclic where the order of A is a power of the prime p. The case |A| = 6 is a special case of the following result. Let |A| = m and |B| = n, where m and n are relatively prime.
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- Groups '93 Galway/St Andrews , pp. 524 - 527Publisher: Cambridge University PressPrint publication year: 1995