Exactm-covers of Groups by Cosets

Let G be a group covered by its left cosets a1G1, · · · , akGk exactly m times. It is known that [G : ⋂k i=1 Gi] 6 k!. When all the Gi are subnormal in G and ⋂k i=1 Gi = H, we are able to determine the least value of k in terms of m,G,H. For any i = 1, · · · , k, providing G/(Gi)G is solvable we show that k > m + f([G : Gi]) and hence [G : Gi] 6 2k−m, where f(n) = ∑r s=1 αs(ps − 1) if p1 1 · · · p αr r is the standard factorization of n. These extend some previous results on disjoint covers.