On the norm of the nilpotent residuals of all subgroups of a finite group

On the nilpotent residuals of all subalgebras of Lie algebras

“…This is analogous to the concept of S(G)-subgroup as introduced by Shen, Shiand and Qian (see [12]); this concept has since been further studied by a number of authors, including Gong and Guo (see [7], [8]), Su and Wang (see [15]).…”

On theπF-norm and theH–F-norm of a finite group

“…Recently, Li and Shen [19] considered the intersection of the normalizers of the derived subgroups of all subgroups of G. Also, in [12] and [24], the authors considered the intersection of the normalizers of the nilpotent residuals of all subgroups of G. Furthermore, for a formation F, Su and Wang [29] investigated the intersection of the normalizers of the Fresiduals of all subgroups of G and the intersection of the normalizers of the products of the F-residuals of all subgroups of G and O p ′ (G). As a continuation of the above ideas, we now introduce the notion of H-F-norm as follows: the F-residuals of all subgroups of G and the H-radical of G, that is,…”

ON THEσ-NILPOTENT NORM AND THEσ-NILPOTENT LENGTH OF A FINITE GROUP

“…In the case, when σ = σ 1 (see Remark 1.1(i)), we get from Theorem 1.4 and Corollary 1.5 the following known results. COROLLARY 1.6 (See Proposition 2.4 [10]). For any group G, the subgroup S(G) is soluble.…”

“…Recall that the norm N(G) of G is the intersection of the normalizers of all subgroups of G. This concept was introduced by Baer [8] (see also [9]), and the norm and the generalized norm of a group have been studied by many authors. In particular, in the recent paper [10], the following analog of the subgroup N(G) was introduced:…”

“…The basic properties and some interesting applications of the subgroup S(G) were studied in [10]. Such a subgroup is proved to be particularly useful in the study of some properties of soluble groups.…”