Herzog–Schönheim conjecture, vanishing sums of roots of Unity and convex polygons

Abstract

Let G be a group and H₁,…,H_s be subgroups of G of indices $d_1,\dots ,d_s,$ respectively. In 1974, M. Herzog and J. Schönheim conjectured that if ${\left\{H_i{\alpha }_i\right\}}_{i=1}^{i=s},{\alpha }_i\in G,$ is a coset partition of G, then $d_1,\dots ,d_s$ cannot be pairwise distinct. In this article, we present the conjecture as a problem on vanishing sum of roots of unity and convex polygons and prove some results using this approach.

Keywords:

• Herzog-Schonheim conjecture
• Schreier coset graphs
• free groups

2020 Mathematics Subject classification:

• 20E05
• 20F05
• 20F10
• 20F65

Acknowledgments

I am very grateful to the referee for his/her comments that improved very much the clarity and the readability of the article.