Steiner Coset Partitions of Groups

Abstract A coset partition of a group G is a set partition of G into finitely many left cosets of one or more subgroups. A driving force in this research area is the Herzog–Schönheim Conjecture [15], which states that any nontrivial coset partition of a group contains at least two cosets with the same index. Although many families of groups have been shown to satisfy the conjecture, it remains open. A Steiner coset partition of G , with respect to distinct subgroups $H_1,\dots ,H_r$ , is a coset partition of G that contains exactly one coset of each $H_i$ . In the quest of a more structural version of the Herzog–Schönheim Conjecture, it was shown that there is no Steiner coset partition of G with respect to any $r\geq 2$ subgroups $H_i$ that mutually commute [1]. In this article, we show that this result holds for $r=4$ mutually commuting subgroups provided that G does not have $C_2\times C_2\times C_2$ as a quotient, where $C_2$ is the cyclic group of order $2$ . We further give an explicit construction of Steiner coset partitions of the n -fold direct product $G^*=C_p\times \ldots \times C_p$ for p prime and $n\geq 3$ . This construction lifts to every group extension of $G^*$ .