On the Φ-hypercentral subgroups of finite groups¹

1 This research is supported by the grant of NSFC (Grant # 11871062, 11501235), the Natural Science Foundation of the Jiangsu Province (BK20181451), the Key Natural Science Foundation of the Anhui Education Commission (KJ2017A569), the Scientific Research Foundation of the Sichuan Provincial Education Department (18ZA0434), and the Fundamental Research Funds of the China West Normal University (17E091).

Abstract

Assume that is a class of finite groups. A normal subgroup E is Φ- hypercentral in G if E ≤ Z_Φ (G), where Z_Φ (G) denotes the Φ-hypercentre of G. We call a subgroup H is _p-embedded in G, if there exists a p-nilpotent subgroup B of G such that H_p ∈ Syl_p(B) and B is _p-supplemented in G, where H_p is a Sylow p-subgroup of H. In this paper, the main result is that: Let E be a normal subgroup of G. For all p ∈ π(F*(E)) and every noncyclic Sylow p-subgroup P of F*(E), if there is a prime power p^α such that 1 < p^α ≤ | P | and every subgroup H of P with | H| = p^α is _p-embedded in G, then E is Φ-hypercentral in G.

Mathematics Subject Classification (2010):

• 20D10
• 20D20

Key words:

• Φ-hypercentral
• _p-embedded
• Sylow subgroups
• formation

Notes

1 This research is supported by the grant of NSFC (Grant # 11871062, 11501235), the Natural Science Foundation of the Jiangsu Province (BK20181451), the Key Natural Science Foundation of the Anhui Education Commission (KJ2017A569), the Scientific Research Foundation of the Sichuan Provincial Education Department (18ZA0434), and the Fundamental Research Funds of the China West Normal University (17E091).