Claims

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1986_09_01_berger_felzenbaum_fraenkel: Corollary IV of the Canad. Math. Bull. paper (1986): a coset partition of a finite nilpotent group into two or more cosets has two cosets of the same order; accepted on the refereed paper.

2003_06_05_sun: Theorem 1.1 of Sun's J. Algebra paper (2004): a nontrivial uniform cover by cosets of subnormal subgroups repeats an index, so no abelian group has a counterexample; accepted on the refereed paper.

2018_03_09_margolis_schnabel: Theorem A of Margolis and Schnabel's Beitr. Algebra Geom. paper (2019): every group of order below 1440 satisfies the Herzog-Schönheim conjecture; accepted on the refereed paper.

2024_04_25_akman_sissokho: Akman and Sissokho's Beitr. Algebra Geom. paper (2025): a coset partition of any group into cosets of two to seven distinct subgroups repeats an index, so a counterexample needs eight cells; accepted on the refereed paper.

2026_08_17_itabe: Rio Itabe's unreviewed computer-assisted proof, with a Lean 4 release, that every exact partition of a group into two to seventeen cosets of finite-index subgroups repeats an index; the general question stays open.

2026_10_02_menon: Murali Menon's AI-assisted Lean 4 proof, registered with Palomar, that a partition of a solvable group into two or more cosets repeats an index; unreviewed, with no informal write-up; pending.

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