This research proves the Herzog-Schönheim conjecture for simple groups and symmetric groups, indicating subgroup index relationships.
The Herzog-Schönheim conjecture states that if H₁, …, Hₖ are subgroups of a group G and x₁, …, xₖ are elements of G such that H₁x₁, …, Hₖxₖ 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.
No takes yet. Share an insight, caveat, or question.
Garonzi et al. (2025) studied this question.
Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context: