Pronormal Subgroups Research Articles

Some results on variations on the norm of finite groups

AbstractLet G be a finite group and $$N_{\Omega }(G)$$ N Ω ( G ) be the intersection of the normalizers of all subgroups belonging to the set $$\Omega (G),$$ Ω ( G ) , where $$\Omega (G)$$ Ω ( G ) is a set of all subgroups of G which have some theoretical group property. In this paper, we show that $$N_{\Omega }(G)= Z_{\infty }(G)$$ N Ω ( G ) = Z ∞ ( G ) if $$\Omega (G)$$ Ω ( G ) is one of the following: (i) the set of all self-normalizing subgroups of G; (ii) the set of all subgroups of G satisfying the subnormalizer condition in G; (iii) the set of all pronormal subgroups of G; (iv) the set of all weakly normal subgroups of G; (v) the set of all NE-subgroups of G.

Pronormality in uncountable groups

A subgroup X of a group G is called pronormal if X and X g are conjugate in 〈 X , X g 〉 for each element g of G. Pronormal subgroups play a relevant role in many problems of group theory; for instance, they arise naturally in the study of groups in which normality is a transitive relation, because a subgroup is normal if and only if it is subnormal and pronormal. Groups with only pronormal subgroups have been described by Kuzennyĭ and Subbotin (at least in the locally graded case) and the aim of this paper is to investigate the structure of uncountable locally graded groups of cardinality ℵ in which all subgroups of cardinality ℵ are pronormal.

On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups

On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups

The influence of weakly pronormal subgroups on the supersolvability of finite groups

The influence of weakly pronormal subgroups on the supersolvability of finite groups

LOCALLY FINITE SIMPLE GROUPS WHOSE NONNILPOTENT SUBGROUPS ARE PRONORMAL

AbstractA subgroup H of a group G is said to be pronormal in G if each of its conjugates $H^g$ in G is already conjugate to it in the subgroup $\langle H,H^g\rangle $ . The aim of this paper is to classify those (locally) finite simple groups which have only nilpotent or pronormal subgroups.

The influence of weakly pronormal subgroups on the structure of finite groups

Let G be a finite group. A subgroup H of G is called pronormal in G if for every H and H are conjugate in A subgroup H of G is called weakly pronormal in G if there exists a subgroup K of G such that G = HK and is pronormal in G. In this paper, we investigate the structure of G under the assumption that some subgroups of G are weakly pronormal in G.

Groups with pronormal deviation

The structure of locally soluble groups in which all subgroups are pronormal is known, and many authors have investigated groups which are rich in pronormal subgroups. Here we prove that if a radical group G has pronormal deviation, which means that the set of its non-pronormal subgroups satisfies a very weak chain condition, then either G is minimax or all its subgroups are pronormal. It follows that if a radical group has pronormal deviation, then its pronormal deviation is at most 1.

On the Pronormality of Second Maximal Subgroups in Finite Groups with Socle $$L_{2}(q)$$

According to P. Hall, a subgroup \(H\) of a finite group \(G\) is called pronormal in \(G\) if, for any element \(g\) of \(G\), the subgroups \(H\) and \(H^{g}\) are conjugate in \(\langle H,H^{g}\rangle\). The simplest examples of pronormal subgroups of finite groups are normal subgroups, maximal subgroups, and Sylow subgroups. Pronormal subgroups of finite groups were studied by a number of authors. For example, Legovini (1981) studied finite groups in which every subgroup is subnormal or pronormal. Later, Li and Zhang (2013) described the structure of a finite group \(G\) in which, for a second maximal subgroup \(H\), its index in \(\langle H,H^{g}\rangle\) does not contain squares for any \(g\) from \(G\). A number of papers by Kondrat’ev, Maslova, Revin, and Vdovin (2012–2019) are devoted to studying the pronormality of subgroups in a finite simple nonabelian group and, in particular, the existence of a nonpronormal subgroup of odd index in a finite simple nonabelian group. In The Kourovka Notebook, the author formulated Question 19.109 on the equivalence in a finite simple nonabelian group of the condition of pronormality of its second maximal subgroups and the condition of Hallness of its maximal subgroups. Tyutyanov gave a counterexample \(L_{2}(2^{11})\) to this question. In the present paper, we provide necessary and sufficient conditions for the pronormality of second maximal subgroups in the group \(L_{2}(q)\). In addition, for \(q\leq 11\), we find the finite almost simple groups with socle \(L_{2}(q)\) in which all second maximal subgroups are pronormal.

On some types of antinormal subgroups

A subgroup $H$ is called a pronormal subgroup in a group $G$ if for any $g \in G$ subgroups $H$ and $H^g$ are conjugated in ${<}H, H^g{>}$. A subgroup $H$ is called a contranormal subgroup in a group $G$ if $H^G = G$. A subgroup $H$ is called a weak pronormal subgroup in a group $G$ if $N_K(H)$ is contranormal in any subgroup $K$ which contains $H$. We obtained the examples of non pronormal but weak pronormal subgroups.

Structure of finite groups, in which any pronormal subgroup is either normal or abnormal

A subgroup $H$ of a group $G$ is said to be abnormal in $G$ if, for each element $g \in G$, we have $g \in {<}H, H^g{>}$. A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if, for each element $g \in G$, the subgroups $H$ and $H^g$ are conjugate in ${<}H, H^g{>}$. We describe all finite groups, each pronormal subgroup in which is either normal or abnormal.

Finite groups with some weakly pronormal subgroups

AbstractA subgroupHof a finite groupGis called weakly pronormal inGif there exists a subgroupKofGsuch thatG=HKG=HKandH∩KH\cap Kis pronormal inG. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

A note on pronormal p-subgroups of finite groups

In this note, we investigate the influence of pronormal p-subgroups on the structure of finite groups. We not only simplify, but also generalize some main results of Liu et al. (J Algebra Appl 19:2050110, 2020), Shen et al. (Monatshefte Math 175:629–638, 2014) and Yu and Lai (Monatshefte Math 181:745–747, 2016). We also prove that for p-subgroups of finite groups, the concept of pronormal subgroups, weakly normal subgroups and weakly closed subgroups are equivalent.

English

For a finite group G and a fixed Sylow p-subgroup P of G, Ballester-Bolinches and Guo proved in 2000 that G is p-nilpotent if every element of P ∩ G′ with order p lies in the center of NG(P) and when p = 2, either every element of P ∩ G′ with order 4 lies in the center of NG(P) or P is quaternion-free and NG(P) is 2-nilpotent. Asaad introduced weakly pronormal subgroup of G in 2014 and proved that G is p-nilpotent if every element of P with order p is weakly pronormal in G and when p = 2, every element of P with order 4 is also weakly pronormal in G. These results generalized famous Ito’s Lemma. We are motivated to generalize Ballester-Bolinches and Guo’s Theorem and Asaad’s Theorem. It is proved that if p is the smallest prime dividing the order of a group G and P, a Sylow p-subgroup of G, then G is p-nilpotent if G is S4-free and every subgroup of order p in P ∩ Px ∩ $$G^{\mathfrak{N}_{\mathfrak{p}}}$$ is weakly pronormal in NG(P) for all x ∈ GNG(P), and when p = 2, P is quaternion-free, where $$G^{\mathfrak{N}_{\mathfrak{p}}}$$ is the p-nilpotent residual of G.

Finite groups with some pronormal subgroups

A subgroup [Formula: see text] of a finite group [Formula: see text] is called pronormal in [Formula: see text] if for each [Formula: see text], the subgroups [Formula: see text] and [Formula: see text] are conjugate in [Formula: see text]. In this paper, we analyze how certain properties of pronormal subgroups influence the structure of groups.

A generalization to Sylow permutability of pronormal subgroups of finite groups

In this note, we present a new subgroup embedding property that can be considered as an analogue of pronormality in the scope of permutability and Sylow permutability in finite groups. We prove that finite PST-groups, or groups in which Sylow permutability is a transitive relation, can be characterized in terms of this property, in a similar way as T-groups, or groups in which normality is transitive, can be characterized in terms of pronormality.

On Pronormal Subgroups in Finite Simple Groups

A subgroup H of a group G is called pronormal if, for any element g of G, the subgroups H and Hg are conjugate in the subgroup they generate. Some problems in the theory of permutation groups and combinatorics have been solved in terms of pronormality, and the characterization of pronormal subgroups in finite groups is a problem of importance for applications of group theory. A task of special interest is the study of pronormal subgroups in finite simple groups and direct products of such groups. In 2012 E.P. Vdovin and D.O. Revin conjectured that the subgroups of odd index in all finite simple groups are pronormal. We disproved this conjecture in 2016. Accordingly, a natural task is to classify finite simple groups in which the subgroups of odd index are pronormal. This paper completes the description of finite simple groups whose Sylow 2-subgroups contain their centralizers in the group and the subgroups of odd index in which are pronormal.

Про деякі узагальнення пронормальних підгруп

Про деякі узагальнення пронормальних підгруп

On some generalization of pronormal subgroups in locally finite group

We proved that if every subgroup of locally finite group is monopronormal, then this group is a $\overline{T}$-group.

Groups in which normality is a weakly transitive relation

We study groups in which normality is a weakly transitive relation, giving an extension of in Theorem A [On finite T-groups, J. Aust. Math. Soc. 75 (2003) 181–191] due to Ballester-Bolinches and Esteban-Romero pointing out the relations between these groups and those in which all subgroups are almost pronormal. Moreover, we extend a well-known theorem of Peng [Finite groups with pro-normal subgroups, Proc. Amer. Math. Soc. 20 (1969) 232–234] proving that for a large class of generalized FC-groups the weak transitivity of normality is equivalent to having finitely many maximal pronormalizers of subgroups.

ON GENERALISED PRONORMAL SUBGROUPS OF FINITE GROUPS

AbstractFor 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/CoreG(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 = U0 ≤ U1 ≤ . . . ≤ 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.