On the Herzog–Schönheim conjecture for uniform covers of groups

Let G be any group and a1G1,…,akGk (k>1) be left cosets in G. In 1974 Herzog and Schönheim conjectured that if is a partition of G then the (finite) indices n1=[G:G1], …, nk=[G:Gk] cannot be pairwise distinct. In this paper we show that if covers all the elements of G the same number of times and G1,…,Gk are subnormal subgroups of G not all equal to G, then M=max1 j k{1 i k: ni=nj} is not less than the smallest prime divisor of n1 nk; moreover, min1 i klogni=O(Mlog2M) where the O-constant is absolute.