On the Herzog-Sch\"onheim conjecture for uniform covers of groups

Let G be any group and $a_1G_1,...,a_kG_k (k>1)$ be left cosets in G. In 1974 Herzog and Sch\"onheim conjectured that if $\Cal A=\{a_iG_i\}_{i=1}^k$ is a partition of G then the (finite) indices $n_1=[G:G_1],...,n_k=[G:G_k]$ cannot be distinct. In this paper we show that if $\Cal A$ covers all the elements of G the same times and $G_1,...,G_k$ are subnormal subgroups of G not all equal to G, then $M=\max_{1\le j\le k}|\{1\le i\le k:n_i=n_j\}|$ is not less than the smallest prime divisor of $n_1... n_k$, moreover $\min_{1\ls i\ls k}\log n_i=O(M\log^2 M)$ where the O-constant is absolute.