Document Type

Article

Publication Title

Canadian Mathematical Bulletin

Publication Date

6-2025

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, 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 H1,...,Hr, is a coset partition of G that contains exactly one coset of each Hi. 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 ≥ 2 subgroups Hi that mutually commute. In this article, we show that this result holds for r = 4 mutually commuting subgroups provided that G does not have C2 X C2 X C2 as a quotient, where C2 is the cyclic group of order 2. We further give an explicit construction of Steiner coset partitions of the n-fold direct product G∗=Cp X … X Cp for p prime and n ≥ 3. This construction lifts to every group extension of G∗.

Funding Source

This article was published Open Access thanks to a transformative agreement between Milner Library and Cambridge University Press.

Comments

First published in Canadian Mathematical Bulletin (2025): https://doi.org/10.4153/S0008439525100787

Published by Cambridge University Press on behalf of Canadian Mathematical Society. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.

DOI

10.4153/S0008439525100787

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