Weak second maximal subgroups in solvable groups

In this paper, we investigate the differences between weak second maximal subgroups and second maximal subgroups. A sufficient and necessary condition is also given to describe a class of groups whose weak second maximal subgroups coincide with its second maximal subgroups (called WSM-groups) under the solvable case. As an application, we will prove that every non-vanishing element of a solvable WSM-group lies in its Fitting subgroup.