On c-normality of finite groups

Abstract

A subgroup H of a finite group G is said to be c-normal in G if there exists a normal subgroup N of G such that G = HN with H n N < H a = Core o (//). We are interested in studying the influence of the c-normality of certain subgroups of prime power order on the structure of finite groups.2000 Mathematics subject classification: primary 20D10, 20D30.

On Weakly ℋ-embedded Subgroups of Finite Groups

“…Thus G = PK Since G/K P/ P ∩ K and H n be the set of all maximal subgroups of P By the hypothesis of the theorem, H i is weakly H-embedded in G where i = 1 2 n Then G has a normal subgroup K i such that H G i = H i K i and H i ∩ K i ∈ H G for all i. Consider PK i for all i If PK i < G for some i, then, by (2), PK i is p-nilpotent and so K i is p-nilpotent. By (4), O p G = 1 so K i = 1 and K i is not a p-subgroup of G Then O p G = 1 contradicting (1).…”

“…By (4), O p G = 1 so K i = 1 and K i is not a p-subgroup of G Then O p G = 1 contradicting (1). Thus G = PK i for all i Let T = n i=1 K i Clearly, T is a nontrivial normal subgroup of G Then, by (1), P ∩ T = 1 By Grün's Theorem ([12, Satz 3.4, p 423]), T ∩ P ∩ T =< T ∩ P ∩ N T T ∩ P T ∩ P ∩ T ∩ P g g ∈ T > By (4), N G T ∩ P < G Since T ∩ P is normal in P it follows that P ≤ N G T ∩ P Then, by (2), N G T ∩ P is p-nilpotent and so N T T ∩ P is p- (1) and (4). The proof of the theorem is complete.…”

“…We use conventional notions and notation as in [8]. A subgroup H of a group G is said to be c-normal in G if there exists a normal subgroup K of G such that G = HK and H ∩ K ≤ H G , where H G = Core G H is the largest normal subgroup of G contained in H Since this concept was introduced by [19], many authors have investigated the structure of finite groups using this concept and a lot of research has been given; see, for example [2,5,10,14,16,20,21]. Following [6], a subgroup H of a group G is called an H-subgroup in G if N G H ∩ H g ≤ H for all g ∈ G. They showed that a group G is a supersolvable T -group if and only if every subgroup of G is an H-subgroup in G a T -group is a group in which every subnormal subgroup is normal .…”

Influence of weakly H-subgroups of minimal subgroups on the structure of finite groups

“…Also the class F contains the class U as the commutator subgroup of a supersolvable group is nilpotent. Now consider the group G = GL (2,3). This group has a normal subgroup H isomorphic to the quaternion group of order 8 such that G/H ∼ = S 3 , the symmetric group of order 6, and therefore we have G/H ∈F.…”

“…How minimal subgroups can be embedded in a group G is a question of particular interest in studying the structure of G. In fact, many authors have investigated the influence of normality, c-normality, H-subgroup and more recently weakly H-subgroup of the minimal subgroups of a group G on the structure of G; see for example [1], [3,4], [6,7], [10], [12,20] and [22,23]. The present paper may be viewed as a continuation of [1], [17] and [20].…”