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.