Finite groups with Frobenius normalizer condition for non-normal primary subgroups

2018/06/10 by Chi, Zhang, Guo, Wenbin
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1806.03672

Abstract

A finite group P is said to be primary if |P|=p^a for some prime p. We say a primary subgroup P of a finite group G satisfies the Frobenius normalizer condition in G if N_G(P)/C_G(P) is a p-group provided P is p-group. In this paper, we determine the structure of a finite group G in which every non-subnormal primary subgroup satisfies the Frobenius normalized condition. In particular, we prove that if every non-normal primary subgroup of G satisfies the Frobenius condition, then G/F(G) is cyclic and every maximal non-normal nilpotent subgroup U of G with F(G)U=G is a Carter subgroup of G.

Related

• Finite Groups With Maximal Normalizers I
• Finite solvable groups with a nilpotent normal complement subgroup
• Finite groups with \mathfrakF-subnormal normalizers of Sylow subgroups
• Finite groups with some particular maximal invariant subgroups being nilpotent or all non-nilpotent maximal invariant subgroups being normal
• A group theoretic proof of Frobenius' theorem