Let G be a finite group and H a subgroup of G. H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G and HsG the intersection of all S-quasinormal subgroups of G containing H. The symbol |G|p denotes the order of a Sylow p-subgroup of G. We prove the followingTheorem A. Let G be a finite group and p a prime dividing |G|. Then G is p-supersoluble if and only if for every cyclic subgroup H ofḠ (G) of prime order or order 4 (if p = 2), Ḡhas a normal subgroup T such thatHsḠandH∩T=HsḠ∩T.Theorem B. A soluble finite group G is p-supersoluble if and only if for every 2-maximal subgroup E of G such that Op′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T with cyclic Sylow p-subgroups such that EsG = ET and |E ∩ T|p = |EsG ∩ T|p.Theorem C. A finite group G is p-soluble if for every 2-maximal subgroup E of G such that Op′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T such that EsG = ET and |E ∩ Tp = |EsG ∩ T|p.
Agrawal, R. K., Generalized center and hypercenter of a finite group, Proc. Amer. Math. Soc., 54 (1976), 13–21.
Agrawal R. K., 'Generalized center and hypercenter of a finite group' (1976) 54Proc. Amer. Math. Soc.: 13-21.
Agrawal R. K.Generalized center and hypercenter of a finite groupProc. Amer. Math. Soc.1976541321)| false
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Studia Scientiarum Mathematicarum Hungarica
2021 Volume 58
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