Transversal coset partitions of groups

The Herzog–Schönheim Conjecture states that there is no partition of any group into finitely many cosets where the corresponding subgroups have distinct indices (repeated indices may come from the same subgroup). In a different approach from most existing proofs of the conjecture for families of groups, we study the inner structure of coset partitions and propose a stronger conjecture for certain cases: namely, that whenever a group is partitioned by finitely many cosets of mutually commuting subgroups, at least one subgroup must contribute more than one coset to the partition. This conjecture holds for the cases of two and three subgroups, and with some restrictions, of four subgroups. In addition, we prove that for \(2\le r\le 7\) $2\le r\le 7$ distinct subgroups’ cosets, Herzog–Schönheim holds; for \(5\le r\le 7\) $5\le r\le 7$ , our proof is computational. In terms of partition structure, we show that for any group G and either two arbitrary subgroups \(H_1,H_2\) $H_1,H_2$ or three mutually commuting subgroups \(H_1,H_2,H_3\) $H_1,H_2,H_3$ of G, the only possible partitions of G are obtained via successive decompositions of cosets of products of \(H_i\) $H_i$ ’s into cosets of smaller products, until single cosets are reached (a “standard” construction). However, we prove that non-standard constructions exist for \(r\ge 4\) $r\ge 4$ subgroups. Finally, we explore applications to exact multi-covers of groups by cosets and to subspace partitions.