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Claim. Let $\{a_iG_i\}_{i=1}^k$ be a nontrivial uniform cover of a group $G$, every element covered the same number of times, with indices $n_i=[G:G_i]$, and let $H$ be the largest normal subgroup of $G$ contained in every $G_i$. If every $G_i$ is subnormal in $G$, or if $G/H$ is solvable and its Sylow subgroup for the largest prime divisor of $|G/H|$ is normal, then $n_1,\ldots,n_k$ are not pairwise distinct; the theorem also bounds the least index in terms of the largest multiplicity of an index. This is [[../library/covering_systems/sun_2004_herzog_schonheim_conjecture_uniform_covers/theorem_1_1|Theorem 1.1]] of Z.-W. Sun, On the Herzog-Schönheim conjecture for uniform covers of groups, J. Algebra 273 (2004), no. 1, 153--175, recorded with the paper's quantitative [[../library/covering_systems/sun_2004_herzog_schonheim_conjecture_uniform_covers/theorem_4_3|Theorem 4.3]] on the library's source card. An exact covering is a uniform cover of multiplicity one.

Covers. The cases of Problem 274 in which every subgroup of the covering is subnormal, or in which the solvable normal-Sylow condition above holds: no such exact covering by two or more cosets has pairwise different indices. Every subgroup of an abelian group is normal, so Erdős's abelian question of [Er77c] and [ErGr80] has the answer no, as the site's commentary records. The question for coverings with a nonsubnormal subgroup outside that condition stays open.

Depends on. Nothing in this wiki; the theorem is the paper's own.

Acceptance. Refereed: Journal of Algebra 273 (2004), no. 1, 153--175, doi:10.1016/S0021-8693(03)00526-X, issued March 2004; the arXiv v1 of 2003-06-05 names this page. Not reviewed: the site's commentary credits the theorem, but the site labels the problem OPEN, so the credit is not counted as review. Not formalized: no Lean proof of the theorem is recorded.