Can a group be partitioned into finitely many cosets with pairwise distinct indices? (Erdős #274)
Statement
If $G$ is a group, can there exist an exact covering of $G$ by more than one coset, all of different sizes? Precisely: do there exist a group $G$ and finitely many cosets $a_1G_1,\ldots,a_kG_k$ of subgroups $G_i\leq G$, with $k\geq 2$ and the indices $[G:G_i]$ pairwise distinct, such that every element of $G$ lies in exactly one of the cosets? The Herzog–Schönheim conjecture asserts that no group admits such a partition.
Acceptance. FULLY RESOLVES: a proof of the Herzog–Schönheim conjecture — no group can be partitioned into $k\geq 2$ cosets of subgroups with pairwise distinct indices — machine-checkable (Lean/Coq) preferred, else a complete written proof; OR an explicit counterexample: a concretely specified group $G$ together with cosets $a_1G_1,\ldots,a_kG_k$ ($k\geq 2$) of pairwise distinct indices and a machine-checkable verification that they partition $G$ (for finite $G$ this is a finite check a program can perform). ADVANCES: an exhaustive, reproducible verification of the conjecture for all groups of order up to a bound strictly beyond the order-$<1440$ record stated in the background (code plus a certificate of exhaustiveness over the orders covered); a proof for a class of groups strictly containing the subnormal case settled by Sun; or nontrivial structural constraints on a minimal counterexample, with proof. Deliver the proof file, or the witness plus checker, or the verification code with the attained order bound.
Background
Raised by Erdős in [Er77c, p.49], [ErGr80, p.26] and [Er97c, p.53] — where the abelian case was the one asked — and listed as open on erdosproblems.com/274 (fetched 2026-07-13, status 'open', tagged 'group theory | covering systems'; page last edited 31 October 2025). It is a question of Herzog and Schönheim, who conjectured that in any (not necessarily finite) group $G$, finitely many cosets of subgroups with pairwise distinct indices can never partition $G$ — the group-theoretic analogue of exact covering systems of the integers. Known results: Sun [Su04] proved the conjecture whenever all the subgroups $G_i$ are subnormal in $G$; in particular the answer is 'no' for abelian $G$, settling the case Erdős originally asked. Margolis and Schnabel [MaSc19] verified the conjecture for every group of order less than $1440$, and Garonzi and Margolis (2025) have since proved it for finite simple groups and for symmetric groups. The statement has been formalized in Lean in the DeepMind formal-conjectures repository. Venue-adjacent (distinct questions): the integer covering-system problems Erdős #7 (all moduli odd) and Erdős #273 (moduli of the form $p-1$). The attacker's tools: computational group theory — a GAP-style exhaustive verification extending the order-$<1440$ record with reproducible scripts (some orders, e.g. $1536=2^9\cdot 3$ with its enormous group count, are genuine walls and likely need structural pruning such as Sun's subnormal-case theorem) — alongside structural group theory enlarging the class of groups for which the conjecture is proven.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #274 (T. F. Bloom) | website |
| REF-02 | Herzog–Schönheim conjecture (Wikipedia) | website |
| REF-03 | Formalized statement of Erdős #274 (DeepMind formal-conjectures, Lean) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.