Remarks on the Supersolvability of a Group with Prime Indices of Some Subgroups

In the paper, a characterization is obtained for a finite group such that, for each prime p , every maximal subgroup of any Sylow p -subgroup of this group is contained in a subgroup of index p ; in particular, such groups are supersolvable. It is proved that a group G is supersolvable if and only if, for every prime p ∈ π( G ), there is a supersolvable subgroup of index p . New properties of groups containing two supersolvable subgroups of different prime indices are established.