Finite groups whose centralizers of non-central elements are second maximal.

A proper subgroup H of a finite group G is called a second maximal subgroup of G if H is a maximal subgroup of every maximal subgroup M in G with H ≤ M. In this paper, we investigate the structure of the finite group G in which CG(x) is a second maximal subgroup for every non-central element x in G, and prove that either G is solvable or G/Z(G)≅PSL(2,rn), where r = 2 with n a prime or r = 3 with n an odd prime. [ABSTRACT FROM AUTHOR]