Erdős #274 · group theory / covering systems
If is a group then can there exist an exact covering of by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)
Problem statement
If is a group then can there exist an exact covering of by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)
Lean 4
Formal statement
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- Upstream Lean 4 statement for Erdős #274 ↗
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- Formal Conjectures — Erdős Problem 274 ↗
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Background
A question of Herzog and Schönheim, who conjectured more generally that if is any (not necessarily finite) group and are finitely many cosets of subgroups of with distinct indices then the cannot form a partition of .
This conjecture was proved in the case when all the are subnormal in by Sun [Su04]. In particular if is abelian (which was the special case asked about in [Er77c] and [ErGr80]) the answer to the original question is no.
Margolis and Schnabel [MaSc19] proved this conjecture for all groups of size .
Progress & source assessment
The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.
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