On {ℱ}_{\mathrm{h}}-\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l} subgroups of finite groups

Abstract

Let G be a finite group, and let ℱ be a formation of finite groups. We say that a subgroup H of G is {ℱ}_{\mathrm{h}}-\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l} in G if there exists a normal subgroup T of G such that HT is a normal Hall subgroup of G and (H ∩ T )H_G/H_G is contained in the ℱ-hypercenter Z_{\infty }^{ℱ}(G/H_G) of G/H_G. In this paper, we obtain some results about the {ℱ}_{\mathrm{h}}-\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l} subgroups and then use them to study the structure of finite groups.

Key words： Finite group; {ℱ}_{\mathrm{h}}-\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l} subgroup; Sylow subgroup; maximal subgroup; 2-maximal subgroup

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