Groups with a maximal irredundant 6-cover

Abstract

A cover for a group G is a collection of proper subgroups whose union is the whole group G. A cover is irredundant if no proper sub-collection is also a cover, and is called maximal if all its members are maximal subgroups. For an integer n > 2, a cover with n members is called an n-cover. Also, we denote σ(G) = n if G has an n-cover and does not have any m-cover for each integer m > n. In this article, we completely characterize groups with a maximal irredundant 6-cover with core-free intersection. As an application of this result, we characterize the groups G with σ(G) = 6. The intersection of an irredundant n-cover is known to have index bounded by a function of n, though in general the precise bound is not known. We also prove that the exact bound is 36 when n is 6.