Narrow Results

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be an $\mathcal{\mathscr{H}}$-subgroup of $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G$. A subgroup $H$ of $G$ is called weakly $\mathcal{\mathscr{H}}$-subgroup in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is an $\mathcal{\mathscr{H}}$-subgroup in $G$. The aim of this paper is to analyze the structure of a finite group in which the fewer maximal subgroups of Sylow $p$-subgroup are weakly $\mathcal{\mathscr{H}}$-subgroups. Our results improve and generalize many known results.

Let H be a subgroup of a group G. A subgroup H of G is said to be S-quasinormally embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. We say that H is weakly S-quasinormally embedded in G if there exists a normal subgroup T of G such that HT ⊴ G and H ∩ T is S-quasinormally embedded in G. In this paper, we investigate further the influence of weakly S-quasinormally embedded subgroups on the structure of finite groups. A series of known results are generalized.

A subgroup H of a finite group G is called ss-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is s-quasinormal in K. In this paper, we investigate the influence of ss-supplemented minimal subgroups on the structure of finite groups and obtain some interesting results.

Let $H$ be a subgroup of a finite group $G$. We say that $H$ is a $c^{\#}$-normal subgroup of $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is a $\mathrm{\text{CAP}}$-subgroup of $G$. In the present paper, we use $c^{\#}$-normality of subgroups to characterize the structure of finite groups, and establish some necessary and sufficient conditions for a finite group to be $p$-supersolvable, $p$-nilpotent and solvable. Our results extend and improve some recent ones.

Let $G$ be a finite solvable group, and let $p$ be a prime. We obtain some conditions for $G$ to be $p$-nilpotent or $p$-closed in terms of irreducible monomial characters.

In this paper, we prove that a finite group $G$ is nilpotent if and only if for every $p\in \tau (G)$ the normalizer of each Sylow $p$-subgroup of $G$ is $p$-nilpotent, where $\tau (G)=\{p$ is a prime $|$ there exists some maximal subgroup $M$ of $G$ such that $p$ divides $|G:M|\}$.

Let H, C and D be subgroups of a finite group G. The pair (C, D) is called a θ*-pair for H if (1) D = (C ∩ H)_G, 〈C^g, H〉 = G for every g ∈ G, and (2) KH < G for every subgroup K of G with K/D ᐊ G/D and K/D < C/D. In this paper, we obtain some results on θ*-pairs which imply G to be solvable, supersolvable or nilpotent.

We use local formations to study the finite groups in which some of its subgroups are c-normal and f-hypercentral. The p-nilpotency and solvability of finite groups are studied. Some new results are obtained and some known results of p-nilpotent groups and solvable groups are generalized.

We investigate the influences of lengths of conjugacy classes of finite groups on the structure of finite groups. Some sufficient conditions for a finite group to be p-nilpotent and supersolvable are obtained. Some known results are generalized.