The Herzog-Schonheim conjecture for finitely generated groups

Abstract

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. Schönheim 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. We consider the Herzog-Schönheim conjecture for free groups of finite rank and develop a new combinatorial approach, using covering spaces. We define $Y$ the space of coset partitions of $F_n$ and show $Y$ is a metric space with interesting properties. We give some sufficient conditions on the coset partition that ensure the conjecture is satisfied and moreover has a neighborhood $U$ in $Y$ such that all the partitions in $U$ satisfy also the conjecture.

Updated version

Cited by in corpus (2)

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