An approach to the Herzog-Sch\"onheim conjecture using automata

Let $G$ be a group and $H_1$,...,$H_s$ be subgroups of $G$ of indices $d_1$,...,$d_s$ respectively. In 1974, M. Herzog and J. Sch\"onheim conjectured that if $\{H_i\alpha_i\}_{i=1}^{i=s}$, $\alpha_i\in G$, is a coset partition of $G$, then $d_1$,..,$d_s$ cannot be distinct. In this paper, we present a new approach to the Herzog-Sch\"onheim conjecture based on automata and present a translation of the conjecture as a problem on automata.