On a conjecture of Herzer concerning linear partitions (Q1087643)

A partition \(\pi\) (G) of a finite group G is a set of non-identity subgroups such that each non-identity element of G lies in exactly one of the subgroups. A partition is nontrivial if it contains at least two subgroups and is called linear if there is a nonempty set \(\epsilon\) (G) of subgroups of G such that (i) each element of \(\epsilon\) (G) is the union of at least two different elements of \(\pi\) (G), (ii) any two different elements of \(\pi\) (G) are contained in a unique element of \(\epsilon\) (G), (iii) \(G\not\in \epsilon (G).\) The author answers a conjecture of A. Herzer by proving that a finite group with a linear partition is either an abelian p-group, or a Frobenius group. The proof depends on the classification of finite groups with non-trivial partitions.