The Herzog-Schönheim conjecture asserts that if a group is partitioned into finitely many left cosets
of proper subgroups of finite subgroup
index, with , then the indices cannot all be distinct (Herzog and Schönheim
1974, Sun 2004). The cosets must cover the group
exactly once.
The conjecture remains open for arbitrary groups (Bloom 2026). For infinite groups, equality of two indices concerns
the numbers of cosets, rather than the possibly infinite
sizes of the subgroups.
Bloom, T. F. "Erdős Problem 274." Accessed Oct. 26, 2026. https://www.erdosproblems.com/274.Herzog,
M. and Schönheim, J. "Research Problem No. 9." Canad. Math. Bull.17,
150, 1974.Sun, Z.-W. "On the Herzog-Schönheim Conjecture for
Uniform Covers of Groups." J. Algebra273, 153-175, 2004. https://doi.org/10.1016/S0021-8693(03)00526-X.
of proper subgroups of finite subgroup
index, with 
, then the indices ![[G:H_1],...,[G:H_k]](/images/equations/Herzog-SchonheimConjecture/Inline3.svg)
cannot all be distinct (Herzog and Schönheim
1974, Sun 2004). The cosets must cover the group
exactly once.
