Narrow Results

We study the variety of algebras A over a field of characteristic ≠ 2, 3, 5 satisfying the identities xy=yx and β {((xx)y)x-((yx)x)x} + γ {((xx)x)y-((yx)x)x}=0, where β, γ are scalars. We do not assume power-associativity. We prove that if A admits a non-degenerate trace form, then A is a Jordan algebra. We also prove that if A is finite-dimensional and solvable, then it is nilpotent. We find three conditions, any of which implies that a finite-dimensional right-nilalgebra A is nilpotent.

We analyze symplectic forms on six-dimensional real solvable and non-nilpotent Lie algebras. More precisely, we obtain all those algebras endowed with a symplectic form that decompose as the direct sum of two ideals or are indecomposable solvable algebras with a four-dimensional nilradical.

We obtain some sufficient conditions on the number of non-(sub)normal non-abelian subgroups of a finite group to be solvable, which extend a result of Shi and Zhang in 2011.

A subgroup H of G is called ℳ-supplemented in G if there exists a subgroup B of G such that G = HB and H_i B < G for every maximal subgroup H_i of H. In this paper, we use ℳ-supplemented subgroups to study the structure of finite groups and obtain some new characterization about solvability and p-supersolvability for a fixed prime p. Some results in the literature are corollaries of our theorems.

We define perfect ideals, near perfect ideals and upper bounded ideals of a finite-dimensional Lie superalgebra, and study the properties of these three kinds of ideals through their relevant sequences. We prove that a Lie superalgebra is solvable if and only if its maximal perfect ideal is zero, or its quotient superalgebra by the maximal perfect ideal is solvable. We also show that a Lie superalgebra is nilpotent if and only if its maximal near perfect ideal is zero. Moreover, we prove that a nilpotent Lie superalgebra has only one upper bounded ideal, which is the nilpotent Lie superalgebra itself.

An element $x$ of a finite group $G$ is said to be primary if the order of $x$ is a prime power. We define $\mathrm{csp2}(G)$ as follows: if $|x^G|$ is a prime power for every primary element $x$ of $G$, where $x^G$ is the conjugacy class of $x$ in $G$, then $\mathrm{csp2}(G)=0$; if there exists a primary element $x$ in $G$ such that $|x^G|$ is divisible by at least two distinct primes, then $\mathrm{csp2}(G)=\max \{|x^G| | x\in G\text{is primary},|x^G|\text{is divisible by at least two distinct primes}\}$. In this paper we discuss the influence of the number $\mathrm{csp2}(G)$ on the structure of $G$.