(untitled) (Communications in Mathematical Research, 2021)
A finite group G is called a generalized PST-group if every subgroup contained in F(G) permutes all Sylow subgroups of G, where F(G) is the Fitting subgroup of G. The class of generalized PST-groups is not subgroup and quotient group closed, and it properly contains the class of PST-groups. In this paper, the structure of generalized PST-groups is first investigated. Then, with its help, groups whose every subgroup (or every quotient group) is a generalized PST-group are determined, and it is shown that such groups are precisely PST-groups. As applications, T-groups and PT-groups are characterized.