Herzog-Schonheim conjecture, vanishing sums of roots of unity and convex polygons

Abstract

Let $G$ be a group and $H_1$,\ldots,$H_s$ be subgroups of $G$ of indices $d_1,\dots ,d_s$ 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 $d_1,\dots ,d_s$ cannot be distinct. In this paper, we present the conjecture as a problem on vanishing sum of roots of unity and convex polygons and prove some results using this approach.

arXiv admin note: substantial text overlap with arXiv:2001.03882, arXiv:1901.09898

References in corpus (1)

• The Herzog-Schonheim conjecture for finitely generated groups