Narrow Results

A subgroup H of a finite group G is said to be SS-quasinormal in G if there exists a supplement B of H in G such that H is permutable with every Sylow subgroup of B. In this paper, we get some new characterizations of supersolvability and p-nilpotency of G by assuming some subgroups of prime power order of G are SS-quasinormal.

Given a maximal subalgebra M of a finite-dimensional Lie algebra L, a θ-pair for M is a pair (A, B) of subalgebras such that A ≰ M, B is an ideal of L contained in A ∩ M, and A/B includes properly no nonzero ideal of L/B. This is analogous to the concept of θ-pairs associated to maximal subgroups of a finite group, which has been studied by a number of authors. A θ-pair (A, B) for M is said to be maximal if M has no θ-pair (C, D) such that A < C. In this paper, we obtain some properties of maximal θ-pairs and use them to give some characterizations of solvable, supersolvable and nilpotent Lie algebras.

A subgroup H of a group G is called Φ-s-supplemented in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ Φ (H), where Φ(H) is the Frattini subgroup of H. We investigate the influence of Φ-s-supplemented subgroups on the p-nilpotency, p-supersolvability and supersolvability of finite groups.

This paper aims to study the concept of $\varphi$-ideals of a finite-dimensional Lie algebra which is analogous to the concept of $c$-ideal and $c$-normal subgroup. We compile some basic properties of $\varphi$-ideals and consider the influence of this concept on the structure of a finite-dimensional Lie algebra, especially its solvability and supersolvability. We also show that the counterpart of some results for $c$-ideals of Lie algebras and $c$-normal subgroups are not valid here.

Let $G$ be a finite group and $H$ a subgroup of $G$. $H$ is said to be $NH$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is a Hall subgroup of $G$ and $H\cap T\le H_{\bar{s}G}$, where $H_{\bar{s}G}$ is the largest $s$-semipermutable subgroup of $G$ contained in $H$. In this paper, we give some new characterizations of $p$-nilpotent and supersolvable groups by using $NH$-embedded subgroups. Some known results are generalized.

Let $G$ and $A$ be groups where $A$ acts on $G$ by automorphisms. We say “the action of $A$ on $G$ is good” if the equality $H=[H,B]C_H(B)$ holds for any subgroup $B$ of $A$ and for any $B$-invariant subgroup $H$ of $G$. It is straightforward that every coprime action is a good action. In this work, we extend some results due to Ward, Gross, Shumyatsky, Jabara and Meng and Guo under coprime action to good action.

Let G be a finite group, we define the average codegree of the irreducible characters of G as $\mathrm{\text{acod}}(G)=\frac{1}{|\mathrm{\text{Irr}}(G)|}{\sum }_{\chi \in \mathrm{\text{Irr}}(G)}\mathrm{\text{cod}}(\chi )$, where $\mathrm{\text{cod}}(\chi )=\frac{|G:ker\chi |}{\chi (1)}$. We prove that if $G$ is non-solvable, then $\mathrm{\text{acod}}(G)\ge 68/5$, and the equality holds if and only if $G\cong A_5$. Also, we show that if $G$ is non-supersolvable, then $\mathrm{\text{acod}}(G)\ge 11/4$, and the equality holds if and only if $G\cong A_4$. In addition, we obtain that if $p$ is the smallest prime divisor of $\left|G\right|$, then $\mathrm{\text{acod}}(G)<p$ if and only if $G$ is an elementary abelian $p$-group.

We introduce a new subgroup embedding property of finite groups called c-normal embedding. By using this embedding property, formation theory and some special techniques, we obtain `iff' versions of theorems of Itô, Buckley, Srinivasan and other related results on p-nilpotence, nilpotence and supersolvability.

A subgroup H of a finite group G is called an ℋ-subgroup of G if N_G(H) ∩ H^g ≤ H for all g ∈ G. The set of all ℋ-subgroups of a finite group G is denoted by ℋ(G). In this paper, a sufficient condition about p-nilpotency is given and some new results for a finite group G to be p-nilpotent or supersolvable are obtained based on the assumption that some subgroups belong to ℋ(G).