Minimal partition-free groups

TL;DR

This paper investigates finite groups that cannot be partitioned non-trivially, focusing on those where all proper non-cyclic subgroups do admit such partitions, revealing structural properties of these groups.

Contribution

It characterizes partition-free groups with the property that all their proper non-cyclic subgroups are partitionable, advancing understanding of subgroup partition structures.

Findings

Identification of conditions for a group to be partition-free

Characterization of proper non-cyclic subgroups admitting partitions

Insights into the subgroup structure of partition-free groups

Abstract

Let G be a finite group. A collection P={H1, ..., Hr} of subgroups of G, where r > 1, is said a non-trivial partition of G if every non-identity element of G belongs to one and only one Hi, for some 1 <=i<=r. We call a group G that does not admit any non-trivial partition a partition-free group. In this paper, we study a partition-free group G whose all proper non-cyclic subgroups admit non-trivial partitions.