Herzog–Schönheim conjecture

The Herzog–Schönheim conjecture states that every finite partition of a group into cosets of subgroups contains at least two subgroups with the same index.

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Coset: A translate of a subgroup within a group. The conjecture partitions groups into sets of this form.
Lagrange's theorem: A theorem stating that the order of a subgroup of a finite group divides the group's order. It relates subgroup indices to subgroup sizes in finite groups.
Herzog–Schönheim conjecture for finite groups: The Herzog–Schönheim conjecture restricted to partitions of finite groups by cosets of subgroups. Finite groups form a major setting in which the conjecture has been investigated.
Marcel Herzog: An Israeli mathematician known for work in group theory. He formulated the conjecture jointly with Schönheim.
Counterexample: An example that disproves a universal mathematical statement. No counterexample to the conjecture is known, despite extensive searches.
Index of a subgroup: The number of left cosets of a subgroup in a group. The conjecture asks whether two subgroups in a partition must have equal index.
Coset partition identity: An identity expressing the indicator function of a finite coset partition as the sum of its coset indicators. Applying characters to this identity yields arithmetic constraints on subgroup indices.
Herzog–Schönheim conjecture for abelian groups: The Herzog–Schönheim conjecture restricted to groups whose operation is commutative. Commutativity permits stronger character-based arguments than the general setting.
Avi Schönheim: A mathematician whose work included combinatorics and group theory. He formulated the conjecture jointly with Herzog.
Finite-index subgroup: A subgroup having finitely many cosets in its ambient group. Every subgroup in a finite coset partition has finite index.