Formations and Products of F(G)-Subnormal Subgroups of Finite Solvable Groups

A subgroup H of a finite group G is said to be F( G )- subnormal if it is subnormal in H F( G ), where F( G ) is the Fitting subgroup of G. In the paper, the problem of whether or not a formation β contains products of F( G )-subnormal β-subgroups of finite solvable groups is studied. In particular, solvable saturated formations β with this property are described. Formation properties of groups having three solvable F( G )-subnormal subgroups with pairwise coprime indices are studied. The supersolvability of any group G having three supersolvable F( G )-subnormal subgroups whose indices in G are pairwise coprime is proved.