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21.93 (2026)

Open

(M. Herzog, J. Schönheim). Let $G$ be a group and let $k \geqslant 2$. Let $H_1, \dots, H_k$ be subgroups of $G$, and $g_1, \dots, g_k$ elements of $G$ such that the cosets $g_1 H_1, \dots, g_k H_k$ form a partition of $G$. Is it true that $|G : H_i| = |G : H_j|$ for some $i \neq j$?

Contributor: L. Margolis

Progress

This is known to be true for groups with a Sylow tower (M. A. Berger, A. Felzenbaum, A. Fraenkel, Fund. Math., 128, no. 3 (1987), 139–144). Also cf. 20.99.

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