Centralizers in a group whose central factor is simple

Abstract

In this paper, we find the number of the element centralizers of a finite group [Formula: see text] such that the central factor of [Formula: see text] is the projective special linear group of degree 2 or the Suzuki group. Our results generalize some main results of [Ashrafi and Taeri, On finite groups with a certain number of centralizers, J. Appl. Math. Comput. 17 (2005) 217–227; Schmidt, Zentralisatorverbände endlicher Gruppen, Rend. Sem. Mat. Univ. Padova 44 (1970) 97–131; Zarrin, On element centralizers in finite groups, Arch. Math. 93 (2009) 497–503]. Also, we give an application of these results.