On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups

Abstract

Abstract. Let G be a finite group. A subgroup A of G is said to be S-quasinormal in G if AP ¼ PA for all Sylow subgroups P of G. The symbol H sG denotes the subgroup generated by all those subgroups of H which are S-quasinormal in G. A subgroup H is said to be S-supplemented in G if G has a subgroup T such that T V H c H sG and HT ¼ G; see [24].Theorem A. Let E be a normal subgroup of a finite group G. Suppose that for every non-cyclic Sylow subgroup P of E, either all maximal subgroups of P or all cyclic subgroups of P of prime order and order 4 are S-supplemented in G.

On Π𝔉-Embedded SUBGROUPS OF FINITE GROUPS

“…If all subgroups with prime order and order 4 are in Z U (G), then G is supersolvable (B. Huppert, K. Doerk). Recently, by using the F-hypercentre to study the structure of a group, a large number of new results were obtained (see, for example, [1, 3,5,[9][10][11][18][19][20]23]). In connection with this, we naturally ask: what effect does the πF-hypercentre have on the structure of a group?…”

A $${\mathfrak{U}_m}$$ U m -subnormal subgroup of a finite group

“…for every non-cyclic chief factor H/K of G, every automorphism of H/K induced by an element of G is inner) if and only if it has a normal hypercyclically embedded subgroup E such that G/E is semisimple (Guo and Skiba [3,Theorem C]). Also hypercyclically embedded subgroups play important role in the theory of soluble groups (see [2,4,5]) and the conditions under which a normal subgroup is hypercyclically embedded in G were found by many authors (see the monographs [2,4,5], and, for example, the recent papers [6][7][8][9][10][11][12]). …”

On Boundary Factors and Traces of Subgroups of Finite Groups

“…Some results related to such subgroups are discussed in the books [2,4,5]. Among recent papers in this line of researches, see for example [6][7][8][9][10][11][12][13].…”