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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
For a formation , a subgroup M of a finite group G is said to be -pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug 〉 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 -normal in G if G/Core G(M) belongs to . A subgroup U of a finite group G is called K--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 -normal in Ui , for i = 1,2, …, n. We call a finite group G an -group if every K--subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations the structure of -groups. We pay special attention to the -pronormal subgroups in this analysis.
2010 Mathematics subject classification: 20D10; 20D35; 20F17