Nilpotent groups having a maximal irredundant 11-covering with core-free intersection

Let G be a finite group. A covering of G is a collection of proper subgroups of G whose union is equal to the entire G. If the number of proper subgroups in the covering is n, then the covering is called an n-covering. Considering that no group can be covered by two of its proper subgroups, n ≥ 3. An n-covering is called irredundant if no proper sub-collection of subgroups from the covering is able to cover G. If all members of an n-covering are maximal normal subgroups of G, then the covering is called a maximal n-covering. Let D be the intersection of all members of an n-covering. Then, the covering is said to have a core-free intersection if ∩g∈G gDg−1 = {1}. This paper characterizes nilpotent groups having a maximal irredundant 11-covering with a core-free intersection. It was found that a nilpotent group G has a maximal irredundant 11-covering with a core-free intersection if and only if it is isomorphic to (C2)10, (C3)5, (C3)6, (C5)3 or (C5)4.