Groups in which every element centralizer is a TI -subgroup

Abstract

Let G be a finite group. Recall that a subgroup H is called a TI-subgroup of G if \(H ∩ H^g= 1\)  or H for every element g of G. We call a group G a CTI-group if its every element centralizer is a TI-group. Clearly \(S_3, A_5, D_7\) and \(Q_8\) are all CTI-groups. In this paper, we investigate the structure of a CTI -group G and prove that a CTI -group G is a nilpotent group or a Frobenius group whose complement is either cyclic or the direct product of a cyclic group of odd order and \(Q_8\), or \(G \cong PSL(2, 2^n) \) with n > 1.