Problem Statement
If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)
Categories:
Group Theory Covering Systems
Progress
A question of Herzog and Schönheim, who conjectured more generally that if $G$ is any (not necessarily finite) group and $a_1G_1,\ldots,a_kG_k$ are finitely many cosets of subgroups of $G$ with distinct indices $[G:G_i]$ then the $a_iG_i$ cannot form a partition of $G$.This conjecture was proved in the case when all the $G_i$ are subnormal in $G$ by Sun [Su04]. In particular if $G$ 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 $G$ of size $<1440$.
Source: erdosproblems.com/274 | Last verified: January 14, 2026