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

Let G be any group and $a_1{G}_1,\mathrm{.}\mathrm{.}\mathrm{.},a_k{G}_k(k>1)$ be left cosets in G. In 1974 Herzog and Sch"onheim conjectured that if $\text{Cal}A=\{a_i{G}_i{\}}_{i=1}^k$ is a partition of G then the (finite) indices $n_1=[G:G_1],\mathrm{.}\mathrm{.}\mathrm{.},n_k=[G:G_k]$ cannot be distinct. In this paper we show that if $\text{Cal}A$ covers all the elements of G the same times and $G_1,\mathrm{.}\mathrm{.}\mathrm{.},G_k$ are subnormal subgroups of G not all equal to G, then $M={\max }_{1\le j\le k}\mathrm{\mid }\{1\le i\le k:n_i=n_j\}\mathrm{\mid }$ is not less than the smallest prime divisor of $n_1\mathrm{.}\mathrm{.}\mathrm{.}{n}_k$, moreover $\min_{1\ls i\ls k}\log n_i=O(M\log^2 M)$ where the O-constant is absolute.