A note on the number of centralizers in finite AC-groups

Abstract

Let [Formula: see text] be a finite AC-group such that [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] is odd. We prove that if [Formula: see text] has [Formula: see text] centralizers of elements, then [Formula: see text], [Formula: see text] is an even integer, the set of the sizes of the conjugacy classes of [Formula: see text] is [Formula: see text] and [Formula: see text] is a Frobenius group whose Frobenius kernel is an elementary abelian [Formula: see text]-group of order [Formula: see text] and the Frobenius complement is a group of order [Formula: see text].