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arXiv · 2509.25118

The Herzog-Sch\"onheim conjecture for simple and symmetric groups

M. Garonzi·L. Margolis
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Abstract

The Herzog-Sch\"onheim conjecture states that if $H_1, \ldots, H_k$ are subgroups of a group $G$ and $x_1, \ldots, x_k$ are elements of $G$ such that $H_1x_1, \ldots, H_kx_k$ is a partition of $G$ into cosets, then two of these subgroups must have the same index. We prove this conjecture for simple groups and for symmetric groups.

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M. Garonzi, L. Margolis. 2025-09-29. The Herzog-Sch\"onheim conjecture for simple and symmetric groups. https://arxiv.org/abs/2509.25118

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