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.