The Herzog-Schönheim conjecture for finitely generated groups

יום ד', 25/04/2018 - 10:30

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 a_i\}_{i=1}^{i=s} 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 give some sufficient conditions on the coset partition that ensure the conjecture is satisfied. Furthermore, under a certain assumption, we show there is a finite number of cases to study in order to show the conjecture is true for every coset partition. Since every finitely generated  group is a quotient of a free group of finite rank, we show these results  extend to finitely generated groups.