Book contents
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
On the covers of finite groups
Published online by Cambridge University Press: 15 December 2009
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
Summary
Abstract
A cover for a group is a collection of proper subgroups whose union is the whole group. We report some recent results concerning the different covers of a finite group.
Minimal covers
Let G be a group. A cover of G is a collection A = {Ai|1 ≤ i ≤ n} of proper subgroups of G whose union is G. The subgroups in A are called the components of the cover. The cover is irredundant if no proper sub-collection is also a cover. The cover is minimal if no cover of G has fewer than n members. In this case, J. H. E. Cohn [8] defined σ(G) to be this minimal number of subgroups. It is clear that to study σ(G), it is enough to consider covers consisting of maximal subgroups.
A number of results were proved for soluble groups. In particular it was proved in 1997 by Tomkinson in [14] that if G is a finite noncyclic soluble group, then σ(G) = pa + 1, where pa is the order of a particular chief factor of G. In fact, he proves that
Theorem 1.1[14] Let G be a finite soluble group and let H/K be the smallest chief factor of G having more than one complement in G. Then σ(G) = |H/K| + 1.
In his paper he suggested that it might be of interest to investigate σ(G) for families of simple groups.
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- Information
- Groups St Andrews 2001 in Oxford , pp. 395 - 399Publisher: Cambridge University PressPrint publication year: 2003
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