On the Herzog-Schönheim conjecture for uniform covers of groups

Zhi-Wei Sun

paper

On the Herzog-Schönheim conjecture for uniform covers of groups

arXiv:math/0306099

Abstract

Let G be any group and $a_{1} G_{1}, ..., a_{k} G_{k} (k > 1)$ be left cosets in G. In 1974 Herzog and Schönheim 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 $M = max_{1 \leq j \leq k} ∣ {1 \leq i \leq k : n_{i} = n_{j}} ∣$ 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.

22 pages