On normal subgroups with consecutive G-class sizes

Abstract

Let [Formula: see text] be a group and [Formula: see text] be a normal subgroup of [Formula: see text]. If the set [Formula: see text] is composed by consecutive integers, then [Formula: see text] is either nilpotent or a quasi-Frobenius group with abelian kernel and complements. This is a generalization of Theorem 2 of [A. Beltrán, M. J. Felipe and C. G. Shao, [Formula: see text]-divisibility of conjugacy class sizes and normal [Formula: see text]-complements, J. Group Theory 18 (2015) 133–141].