ON A THEOREM OF SRINIVASAN

The following theorem was derived by S. Srinivasan:Finite group G is supersolvable, if every maximal subgroup of every Sylow subgroup of G is normal.In this paper, two extensions of Srinivasan's theorem are proved.Theorem 1 Finite group G is supersolvable, if every maximal subgroup of every Sylow subgroup of G is S-seminormal.A subgroup H of G is called S-seminormal if HK=KH for any Sylow subgroup K of G with (|H|, |K|)=1.Theorem 2 Finite group G is supersolvable, if every Sylow p-subgroup S of G has mD maximal subgroups which are S-quasinormal for each p||G|, whereA subgroup H of G is called S-quasinormal, if HK=KH for every subgroup K of G.