P-nilpotency Of Finite Groups Research Articles

On Hall normally embedded subgroups and the p-nilpotency of finite groups

Let G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure H G of H in G. In this paper, we shall obtain some characterizations about p-nilpotency of G under the assumption that certain some subgroups of G of prime power order are Hall normally embedded in G.

The p-nilpotency of finite groups with some CSS-subgroups

A subgroup [Formula: see text] of a finite group [Formula: see text] is said to be [Formula: see text]-subgroup of [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is [Formula: see text]-quasinormal in [Formula: see text]. In this paper, we investigate the [Formula: see text]-nilpotency of the finite groups under the assumption that there is a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and every subgroup of [Formula: see text] with order [Formula: see text] or [Formula: see text] (whenever [Formula: see text]) is [Formula: see text]-subgroup, where [Formula: see text] is a prime dividing the order of [Formula: see text] and [Formula: see text] is a Sylow [Formula: see text]-subgroup of [Formula: see text]. As an application of the results, a related problem posed by Heliel et al. in [Finite groups with minimal CSS-subgroups, Eur. J. Pure Appl. Math. 14 (2021) 1002–1014] is solved.

On weakly s-semipermutable subgroups and the p-nilpotency of finite groups

A subgroup H of a group G is called weakly s-semipermutable in G if G has a subnormal subgroup T and an s-semipermutable subgroup HssG contained in H such that G = HT and . In this paper, we present several new criteria on p-nilpotency of finite groups with a small quantity of weakly s-semipermutable maximal subgroups of Sylow p-subgroups.

Engel Condition and p-nilpotency of Finite Groups

Engel Condition and p-nilpotency of Finite Groups

On weakly ℋC-embedded subgroups and p-nilpotence of finite groups

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text] for all [Formula: see text]; [Formula: see text] is called weakly [Formula: see text]-embedded in [Formula: see text] if [Formula: see text] has a normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text] where [Formula: see text] is the normal closure of [Formula: see text] in [Formula: see text]. In this paper, we study the [Formula: see text]-nilpotence of a group [Formula: see text] in which every subgroup of order [Formula: see text] of a Sylow [Formula: see text]-subgroup [Formula: see text] with [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text]. Many recent results in the literature related to [Formula: see text]-nilpotence of [Formula: see text] are generalized.

On the sum of character degrees coprime to p and p-nilpotency of finite groups

The well-known Thompson theorem on character degrees states that if a prime p divides the degree of every nonlinear irreducible character of a finite group G , then G is p -nilpotent. In this paper, we give a strengthened version of Thompson's theorem in terms of ∑ χ ∈ Irr p ′ ( G ) χ ( 1 ) and ∑ χ ∈ Irr p ′ ( G ) χ ( 1 ) 2 , where Irr p ′ ( G ) denotes the set of all ordinary irreducible characters of G of degree coprime to p .

New characterizations of p-nilpotency of finite groups

Suppose that [Formula: see text] is a finite group and [Formula: see text] is a subgroup of [Formula: see text]. [Formula: see text] is said to be an [Formula: see text]-quasinormal subgroup of [Formula: see text] if there is a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] permutes with every Sylow subgroup of [Formula: see text]. In this note, we fix in every non-cyclic Sylow subgroup [Formula: see text] of [Formula: see text] some subgroup [Formula: see text] satisfying [Formula: see text] and study the [Formula: see text]-nilpotency of [Formula: see text] under the assumption that every subgroup [Formula: see text] of [Formula: see text] with [Formula: see text] is [Formula: see text]-quasinormal in [Formula: see text]. The Frobenius theorem is generalized.

π-Quasinormally embedded subgroups and p-nilpotency of finite groups

Let P be a nontrivial p-group. A chain of subgroups is called a maximal chain of P provided that In this article, we determine the structure of finite groups having π-quasinormally embedded subgroups of a maximal chain of Sylow subgroups. We obtain new characterizations of finite p-nilpotent and supersolvable groups.

Nearly s-embedded subgroups and the p-nilpotency of finite groups

A subgroup H of a finite group G is said to be nearly s-embedded in G if G has an s-permutable subgroup T and an s-semipermutable subgroup HssG contained in H such that and where HsG is the intersection of all s-permutable subgroups of G containing H. In this article, we investigate the influence of some nearly s-embedded subgroups of Sylow p-subgroups on the p-nilpotency of finite groups. A number of known results are improved and extended.

On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups

Let \(G\) be a finite group and \(P\) be a \(p\)-subgroup of \(G\). If \(P\) is a Sylow subgroup of some normal subgroup of \(G\), then we say that \(P\) is normally embedded in \(G\). Groups with normally embedded maximal subgroups of Sylow \(p\)-subgroup, where \({(|G|, p-1)=1}\), are studied. In particular, the \(p\)-nilpotency of such groups is proved.

Influence of partially τ-embedded subgroups of prime power order in supersolubility and p-nilpotency of finite groups

In this paper, we introduced a new concept of embedded subgroups which belongs to an embedded class of subgroups of finite groups. A subgroup H of a group G is said to be a partially τ-embedded subgroup in G if there exists a normal subgroup K of G such that HK is normal in G and where generated by all those subgroups of H which are partially τ-quasinormal in G. We investigate the influence of some -embedded subgroups with prime power order on the structure of a finite group G. Some new criteria about the p-nilpotency and supersolubility of a finite group were obtained. Our results also generalized some earlier ones about formations.

On weakly ℌ-embedded subgroups and p-nilpotence of finite groups

Abstract Let G be a finite group and H a subgroup of G. We say that H is an ℌ-subgroup of G if NG (H) ∩ Hg ≤ H for all g ∈G; H is called weakly ℌ-embedded in G if G has a normal subgroup K such that HG = HK and H ∩ K is an ℌ-subgroup of G, where HG is the normal clousre of H in G, i. e., HG = 〈Hg|g ∈ G〉. In this paper, we study the p-nilpotence of a group G under the assumption that every subgroup of order d of a Sylow p-subgroup P of G with 1 < d < |P| is weakly ℌ-embedded in G. Many known results related to p-nilpotence of a group G are generalized.

On the 𝑝-closedness and 𝑝-nilpotency of finite groups

Abstract Let acd ⁡ ( G ) {\operatorname{acd}(G)} and acs ⁡ ( G ) {\operatorname{acs}(G)} denote the average irreducible character degree and the average conjugacy class size, respectively, of a finite group G. The object of this paper is to prove that if acd ⁡ ( G ) < 2 ⁢ ( p + 1 ) / ( p + 3 ) \operatorname{acd}(G)<2(p+1)/(p+3) , then G = O p ⁢ ( G ) × O p ′ ⁢ ( G ) G=O_{p}(G)\times O_{{p^{\prime}}}(G) , and that if acs ⁡ ( G ) < 4 ⁢ p / ( p + 3 ) \operatorname{acs}(G)<4p/(p\kern-1.0pt+\kern-1.0pt3) , then G = O p ⁢ ( G ) × O p ′ ⁢ ( G ) G=O_{p}(G)\kern-1.0pt\times\kern-1.0ptO_{{p^{\prime}}}(G) with O p ⁢ ( G ) {O_{p}(G)} abelian, where p is a prime.

On s-conditional permutability of maximal subgroups of Sylow subgroups of finite groups

Let G be a finite group. A subgroup H of G is said to be s-conditionally permutable in G if, for every prime \(p\in \pi (G)\), there exists a Sylow p-subgroup P of G such that \({ HP}={ PH}\). The purpose here is to study the influence of s-conditional permutability of maximal subgroups of Sylow subgroups on the p-nilpotence of finite groups. Some known results are generalized.

On weakly ℋ-subgroups and p-nilpotency of finite groups

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-subgroup in [Formula: see text] if [Formula: see text] for all [Formula: see text]; [Formula: see text] is called weakly [Formula: see text]-subgroup in [Formula: see text] if it has a normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] is an [Formula: see text]-subgroup in [Formula: see text]. In this paper, we present some sufficient conditions for a group to be [Formula: see text]-nilpotent under the assumption that certain subgroups of fixed prime power orders are weakly [Formula: see text]-subgroups in [Formula: see text]. The main results improve and extend new and recent results in the literature.

On the focal subgroup of a saturated fusion system

The influence of the cyclic subgroups of order p or 4 of the focal subgroup of a saturated fusion system F over a p-group S is investigated in this paper. Some criteria for normality of S in F as well as necessary and sufficient conditions for nilpotency of F are given. The resistance of a p-group in which every cyclic subgroup of order p or 4 is normal, and earlier results about p-nilpotence of finite groups and nilpotency of saturated fusion systems are consequences of our study.

A Note on a Result of Zhang et al.

In this note, we give a new brief proof of a theorem in the article “Sylow Normalizers and p-Nilpotence of Finite Groups” by Zhang, Li, and Miao [9].

Sylow Normalizers and p-Nilpotence of Finite Groups

Let G be a finite group, p a prime and P a Sylow p-subgroup of G. The main goal of the paper is to determine sufficient conditions for the p-nilpotence of a finite group with p-nilpotent Sylow normalizers.

Nearly s-semipermutable subgroups and p-nilpotency

We say that a subgroup H of a group G is nearly s-semipermutable in G provided G has an s-permutable subgroup T such that HT is s-permutable in G and H ∩ T ≤ HssG, where HssG is the subgroup of H generated by all those subgroups of H which are s-semipermutable in G. In this paper, we investigate the influence of near s-semipermutability of some subgroups on the p-nilpotency of finite groups.

S-Conditionally Permutable Subgroups and p-Nilpotency of Finite Groups

We study the p-nilpotency of a group such that every maximal subgroup of its Sylow p-subgroups is s-conditionally permutable for some prime p. By using the classification of finite simple groups, we get interesting new results and generalize some earlier results.