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Paper «On generalised pronormal subgroups of finite groups» published in Glasgow Math. J.

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A. Ballester-Bolinches, J. C. Beidleman, A. D. Feldman, M. F. Ragland

On generalised pronormal subgroups of finite groups

Glasgow Math. J., 56(3) (2014), 691–703

http://dx.doi.org/10.1017/S0017089514000159

Abstract

For a formation $\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug 〉$\mathfrak F$ such that Ux = Ug . Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/Core G(M) belongs to $\mathfrak F$. A subgroup U of a finite group G is called K-$\mathfrak F$-subnormal in G if either U = G or there exist subgroups U = U 0 ≤ U 1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$-normal in Ui , for i = 1,2, …, n. We call a finite group G an $fT_{\mathfrak F}$-group if every K-$\mathfrak F$-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$-groups. We pay special attention to the $\mathfrak F$-pronormal subgroups in this analysis.

2010 Mathematics subject classification: 20D10; 20D35; 20F17