Minimal partition-free groups

Abstract

Let G be a finite group. A collection \(\Pi =\{H_1,\dots ,{H_r}\}\) 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 \(H_i\), for some \(1\leqslant i\leqslant r\). 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.