Narrow Results

In this paper, the concepts of nilpotent and Engel orthomodular lattices, similar with the solvable orthomodular lattices are defined and their properties are investigated. The notion of $n$-Engel orthomodular lattices as a natural generalization of distributive orthomodular lattices is introduced, and we discuss Engel orthomodular lattices, which is defined by left and right normed commutators.

We show every locally solvable subgroup of $\mathrm{\text{PLo}}(I)$ is countable. A corollary is that an uncountable wreath product of copies of $\mathbb{Z}$ with itself does not embed into $\mathrm{\text{PLo}}(I)$.

Let $p$ be a prime and let $ℂ$ be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of $\mathrm{\text{GL}}(p,ℂ)$ up to conjugacy. That is, we give a complete and irredundant list of $\mathrm{\text{GL}}(p,ℂ)$-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of $\mathrm{\text{GL}}(p,ℂ)$ is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For $p\le 3$, we classify all finite irreducible subgroups of $\mathrm{\text{GL}}(p,ℂ)$. Our classifications are available publicly in MAGMA.

We show that fibered 2-knots with closed fiber the Hantzsche–Wendt flat 3-manifold are not reflexive, while every fibered 2-knot with closed fiber a Nil-manifold with base orbifold S(3, 3, 3) is reflexive. We also determine when the knots are amphicheiral or invertible, and give explicit representatives for the possible meridians (up to automorphisms of the knot group which induce the identity on abelianization) for the groups of all knots in either class. This completes the TOP classification of 2-knots with torsion-free, elementary amenable knot group. In the final section, we show that the only non-trivial doubly null-concordant knots with such groups are those with the group of the 2-twist spin of the knot 9₄₆.

If $X$ is an orientable, strongly minimal $P{D}_4$-complex and ${\pi }_1(X)$ has one end, then it has no nontrivial locally finite normal subgroup. Hence, if $\pi$ is a 2-knot group, then (a) if $\pi$ is virtually solvable, then either $\pi$ has two ends or $\pi \cong \mathrm{\Phi }$, with presentation $〈a,t|ta=a^{2}t〉$, or $\pi$ is torsion-free and polycyclic of Hirsch length 4 (b) either $\pi$ has two ends, or $\pi$ has one end and the center $\zeta \pi$ is torsion-free, or $\pi$ has infinitely many ends and $\zeta \pi$ is finite, and (c) the Hirsch–Plotkin radical $\sqrt{\pi }$ is nilpotent.

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.

In this paper we study the structure of a finite group G admitting a solvable group A of automorphisms of coprime order so that for any x ∈ C_G(A) of prime order or of order 4, every conjugate of x in G is also contained in C_G(A). Under this hypothesis it is proven that the subgroup [G, A] is solvable. Also an upper bound for the nilpotent height of [G, A] in terms of the number of primes dividing the order of A is obtained in the case where A is abelian.

All solvable, indecomposable, finite-dimensional, complex Lie superalgebras $𝔤$ whose first derived ideal lies in its nilradical, and whose nilradical is a Heisenberg Lie superalgebra $𝔥$ associated to a ${\mathbb{Z}}_2$-homogeneous supersymplectic complex vector superspace $V$, are here classified up to isomorphism. It is shown that they are all of the form $𝔤=𝔥\oplus 𝔞$, where $𝔞$ is even and consists of non-${ad}_{𝔤}$-nilpotent elements. All these Lie superalgebras depend on an element $\gamma$ in the dual space ${𝔞}^{\ast }$ and on a pair of linear maps defined on $𝔞$, and taking values in the Lie algebras naturally associated to the even and odd subspaces of $V$; namely, if the supersymplectic form is even, the pair of linear maps defined on $𝔞$ take values in $𝔰𝔭(V_0)$, and $𝔬(V_1)$, respectively, whereas if the supersymplectic form is odd these linear maps take values on $𝔤𝔩(V_0)\simeq 𝔤𝔩(V_1)$. When the supersymplectic form is even, a bilinear, skew-symmetric form defined on $𝔞$ is further needed. Conditions on these building data are given and the isomorphism classes of the resulting Lie superalgebras are described in terms of appropriate group actions.

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.

In this paper, we show that every finite-dimensional Zinbiel algebra over an arbitrary field is nilpotent, extending a previous result that they are solvable by other authors.

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 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 $G$.

Let $L$ be a Lie algebra and $\mathrm{\text{Der}}(L)$ and $\mathrm{\text{IDer}}(L)$ be the set of all derivations and inner derivations of $L$, respectively. A derivation $\alpha$ of a Lie algebra $L$ is pointwise inner if $\alpha (x)\in Ima{d}_x$ for all $x\in L$. The set of all pointwise inner derivations of Lie algebra $L$ denoted by ${\mathrm{\text{Der}}}_c(L)$ form a subalgebra of $\mathrm{\text{Der}}(L)$ containing $\mathrm{\text{IDer}}(L)$. In this paper, we prove that, if $L$ is nilpotent of class $k$ (solvable of length $k$), then ${\mathrm{\text{Der}}}_c(L)$ is nilpotent of class $k-1$ (solvable of length $k$ or $k-l$). We also prove that if ${\mathrm{\text{der}}}_c(\frac{L}{Z_j(L)})$ is nilpotent of class $k$, then ${\mathrm{\text{der}}}_c(L)$ is nilpotent of class at most $k+j$, in which ${\mathrm{\text{der}}}_c(L)=\frac{{\mathrm{\text{Der}}}_c(L)}{\mathrm{\text{IDer}}(L)}$ and $Z_j(L)$ is the $j$th term in the upper central series of $L$.

In this survey we discuss recent progress in the study of groups and Lie rings admitting a Frobenius group of automorphisms F H with kernel F and complement H. Our main objective is the case, where a finite group G is acted on by F H in such a way that the fixed-point subgroup of the Frobenius kernel is trivial: C_G(F) = 1. Recent results of Khukhro, Makarenko and Shumyatsky show that in this situation various properties of the group G such as the order, rank, nilpotency, nilpotent length, exponent, nilpotency class are close to the corresponding properties of the fixed-point subgroup of the Frobenius complement, C_G(H). Special emphasis is placed on Lie-theoretic methods used for bounding the nilpotency class and exponent of G.