On the covers of finite groups

Abstract

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 = { A i |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 ) = p a + 1, where p a 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.