Exactm-covers of Groups by Cosets

Let G be a group covered by its left cosets a1G1,⋯ , akGkexactly m times. It is known that [ G:∩i = 1 ⊇ kGi] ⩽k!. When all the Giare subnormal in G and∩i = 1 ⊇ kGi = 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)Gis solvable we show that k⩾m + f([ G: Gi]) and hence [ G: Gi] ⩽ 2 ⊇ k − m, where f(n) = ∑s = 1 ⊇ rαs(ps − 1) if p1 ⊇ α1⋯pr ⊇ αris the standard factorization of n. These extend some previous results on disjoint covers.