Groups in which the centralizer of any non-central element is maximal

Abstract

Abstract Let x be an element of a finite group G. It is clear that 〈 x 〉 ≤ C G ⁢ ( x ) ≤ G {\langle x\rangle\leq C_{G}(x)\leq G} . For the cases where C G ⁢ ( x ) = G {C_{G}(x)=G} and C G ⁢ ( x ) = 〈 x 〉 {C_{G}(x)=\langle x\rangle} for any element x of G, the structure of G is easily decided. In this paper, we investigate the case where C G ⁢ ( x ) {C_{G}(x)} is maximal in G and obtain the structure of G.