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

A proper subgroup [Formula: see text] of a finite group [Formula: see text] is called a second maximal subgroup of [Formula: see text] if [Formula: see text] is a maximal subgroup of every maximal subgroup [Formula: see text] in [Formula: see text] with [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] in which [Formula: see text] is a second maximal subgroup for every non-central element [Formula: see text] in [Formula: see text], and prove that either [Formula: see text] is solvable or [Formula: see text], where [Formula: see text] with [Formula: see text] a prime or [Formula: see text] with [Formula: see text] an odd prime.