Exactm-covers of Groups by Cosets

AbstractLet 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