Journal of Algebra and Its Applications

By the classical theorem of Weitzenböck the algebra of constants $K{[X_d]}^{\delta }$ of a nonzero locally nilpotent linear derivation $\delta$ of the polynomial algebra $K[X_d]=K[x_1,\dots ,x_d]$ in several variables over a field $K$ of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants $F_d{(𝔙)}^{\delta }$ of a locally nilpotent linear derivation $\delta$ of a finitely generated relatively free algebra $F_d(𝔙)$ in a variety $𝔙$ of unitary associative algebras over $K$. It is known that $F_d{(𝔙)}^{\delta }$ is finitely generated if and only if $𝔙$ satisfies a polynomial identity which does not hold for the algebra $U_2(K)$ of $2\times 2$ upper triangular matrices. Hence the free metabelian associative algebra $F_d=F_d(𝔐)=F_d({𝔑}_2𝔄)=F_d(\mathrm{\text{var}}(U_2(K)))$ is a crucial object to study. We show that the vector space of the constants ${(F_d^{\prime })}^{\delta }$ in the commutator ideal $F_d^{\prime }$ is a finitely generated $K{[U_d,V_d]}^{\delta }$-module, where $\delta$ acts on $U_d$ and $V_d$ in the same way as on $X_d$. For small $d$, we calculate the Hilbert series of ${(F_d^{\prime })}^{\delta }$ and find the generators of the $K{[U_d,V_d]}^{\delta }$-module ${(F_d^{\prime })}^{\delta }$. This gives also an (infinite) set of generators of the algebra $F_d^{\delta }$.

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is an ${\mathscr{H}}$-subgroup in $G$ if $N_G(H)\cap H\le H^g$ for all $g\in G$; $H$ is called weakly ${\mathscr{H}}$-subgroup in $G$ if it has a normal subgroup $K$ such that $G=HK$ and $H\cap K$ is an ${\mathscr{H}}$-subgroup in $G$. In this paper, we present some sufficient conditions for a group to be $p$-nilpotent under the assumption that certain subgroups of fixed prime power orders are weakly ${\mathscr{H}}$-subgroups in $G$. The main results improve and extend new and recent results in the literature.

We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_1}\cdots A^{x_k}$ for $A\le G$ and $x_1,\dots ,x_k\in G$, where $A$ is nilpotent or solvable. We derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group which, for a large class of groups, is linear in the non-solvable length of $G$. We also show that every solvable group $G$ is a product of at most $1+c\mathrm{\text{log}}|G:C|$ conjugates of a Carter subgroup $C$ of $G$, where $c$ is a positive real constant. Finally, using these results we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.

Mason introduced the reflexive property for ideals, and recently this concept was extended to many sorts of subsets in rings. In this note, we restrict the reflexivity to nilpotent elements, and a ring will be said to be RNP if it satisfies this restriction. The structure of RNP rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

A finite group $G$ is called a Schur group if any $S$-ring over $G$ is associated in a natural way with a subgroup of $\mathrm{\text{Sym}}(G)$ that contains all right translations. We prove that the groups ${\mathbb{Z}}_3\times {\mathbb{Z}}_{3^n}$, where $n\ge 1$, are Schur. Modulo previously obtained results, it follows that every noncyclic Schur $p$-group, where $p$ is an odd prime, is isomorphic to ${\mathbb{Z}}_3\times {\mathbb{Z}}_3\times {\mathbb{Z}}_3$ or ${\mathbb{Z}}_3\times {\mathbb{Z}}_{3^n}$, $n\ge 1$.

Let $(u,v,w)$ be the ${𝔽}_{q^i}$-rational point on the Fermat curve $u^{q-1}+v^{q-1}+w^{q-1}=0$ with $3|(q^i-1)$. It has recently been proved that if $i\in \{1,2,3\}$ then each $uvw$ is a cube in ${𝔽}_{q^i}$. It is natural to wonder whether there is a generalization to $i\ge 4$. In this paper, we show that the result cannot be extended to $i\ge 4$ in general and conjecture that each $uvw$ is a cube in ${𝔽}_{q^i}$ if and only if $i\in \{1,2,3\}$.

The least number of products of linear forms that may be added together to obtain a given form is the Chow rank of this form. The Chow rank of a generic form corresponds to the smallest $s$ for which the $s$th secant variety of the Chow variety fills the ambient space. We show that, except for certain known exceptions, this secant variety has the expected dimension for low values of $s$.

In this paper, we describe completely the ${𝒦}$-singular subgroup of an abelian group and a ${𝒦}$-nonsingular abelian group in terms of the basic subgroups of its $p$-components and the quotient group by the torsion part. We also prove that a pure subgroup and a quotient group by a pure subgroup of a ${𝒦}$-nonsingular abelian group are ${𝒦}$-nonsingular and give a condition under which a pure extension of a ${𝒦}$-nonsingular abelian group by a ${𝒦}$-nonsingular group is ${𝒦}$-nonsingular.

We characterise, in pointed regular categories, the ideals as the zero-classes of surjective relations. Moreover, we study a variation of the Smith is Huq condition: two surjective left split relations commute as soon as their zero-classes commute.

Suppose the underlying field is of characteristic $p>3$. In this paper, we prove that the automorphisms of the finite-dimensional graded (non-restircited) Lie superalgebras of Cartan-type $W,$ $S,$ $H$ and $K$ can uniquely extend to the ones of the infinite-dimensional Lie superalgebra of Cartan-type $W$. Then a concrete group embedding from $\mathrm{\text{Aut}}X$ into $\mathrm{\text{Aut}}{𝒪}$ is established, where $X$ is any finite-dimensional Lie superalgebra of Cartan-type $W,S,H$ or $K,$ and ${𝒪}$ is the underlying (associative) superalgebra of $W$. The normal series of the automorphism groups of $X$ are also considered.

Let $G$ be a finite group and $n$ any divisor of $\left|G\right|$, the order of $G$. Let $F_n(G)=\{x\in G|x^n=1\}$, Frobenius’ theorem states that $|F_n(G)|=f_n\cdot n$ for some positive integer $f_n$. We call $f_n$ a Frobenius quotient of $G$ for $n$. Let ${ℱ}(G)$ be the set of all Frobenius quotients of $G$, we call ${ℱ}(G)$ the Frobenius spectrum of $G$. In this paper, we give a complete classification of finite groups $G$ with ${ℱ}(G)=\{1,p\}$ for $p$ being the smallest prime divisor of $\left|G\right|$. Moreover, let $G$ be a finite group of even order, ${{ℱ}}^{\prime }(G)$ the set of all Frobenius quotients of $G$ for even divisors of $\left|G\right|$ and $\mathrm{\text{mfe}}(G)$ the maximum Frobenius quotient in ${{ℱ}}^{\prime }(G)$, we prove that $G$ is always solvable if $\mathrm{\text{mfe}}(G)\le 7$ or $\mathrm{\text{mfe}}(G)=8$ and $A_5$ is not a composition factor of $G$.

In this paper, we generalize the main results of Guo and Isaacs, Archiv der Mathematik 105 (2015) 215–222. We also obtain some criteria for $p$-nilpotency of a finite group.

Let $\widehat{𝔤}$ be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable $\widehat{𝔤}$-module is the finite-dimensional irreducible $𝔤$-module, where the action of the simple Lie algebra $𝔤$ is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level $1$ Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra $U_q({\widehat{𝔰𝔩}}_{n+1})$. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator ${{𝒴}}_i(z)$, we obtain an infinite-dimensional vector space $〈{{𝒴}}_i(z)〉$. Next, we introduce an associative algebra $U_q{({𝔰𝔩}_{n+1})}_z$, a certain quantum analogue of the universal enveloping algebra $U({𝔰𝔩}_{n+1})$, and construct some infinite-dimensional $U_q{({𝔰𝔩}_{n+1})}_z$-modules $L{({\lambda }_i)}_z$ corresponding to the finite-dimensional irreducible $U_q({𝔰𝔩}_{n+1})$-modules $L({\lambda }_i)$. We show that the space $〈{{𝒴}}_i(z)〉$ carries a structure of an $U_q{({𝔰𝔩}_{n+1})}_z$-module and, furthermore, we prove that the $U_q{({𝔰𝔩}_{n+1})}_z$-module $〈{{𝒴}}_i(z)〉$ is isomorphic to the $U_q{({𝔰𝔩}_{n+1})}_z$-module $L{({\lambda }_i)}_z$.

The number of minimal nonabelian subgroups in a nonabelian finite $p$-group containing a maximal normal abelian subgroup of given index is estimated. Also the $p$-group for which the above estimate is attained, are described in great detail.

Applying Blackburn’s classification of minimal nonmetacyclic $p$-groups (see Lemma 2(a)), we prove that if $k>1$ and all ${{𝒜}}_k$-subgroups of a finite $p$-group $G$ are metacyclic so is $G$.

In this paper, we continue to study zero-divisor properties of skew polynomial rings $R[x;\alpha ,\delta ]$, where $R$ is an associative ring equipped with an endomorphism $\alpha$ and an $\alpha$-derivation $\delta$. For an associative ring $R$, the undirected zero-divisor graph of $R$ is the graph $\mathrm{\Gamma }(R)$ such that the vertices of $\mathrm{\Gamma }(R)$ are all the nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are connected by an edge if and only if $xy=0$ or $yx=0$. As an application of reversible rings, we investigate the interplay between the ring-theoretical properties of a skew polynomial ring $R[x;\alpha ,\delta ]$ and the graph-theoretical properties of its zero-divisor graph $\mathrm{\Gamma }(R[x;\alpha ,\delta ])$. Our goal in this paper is to give a characterization of the possible diameters of $(\mathrm{\Gamma }(R[x;\alpha ,\delta ]))$ in terms of the diameter of $\mathrm{\Gamma }(R)$, when the base ring $R$ is reversible and also have the $(\alpha ,\delta )$-compatible property. We also completely describe the associative rings all of whose zero-divisor graphs of skew polynomials are complete.

Following the prototype of dimonoids, diquasigroups are directed versions of quasigroups, where the structure is split into left and right quasigroups on the same set. The linear and affine diquasigroups that form the topic of this paper are built on the foundation of a module. In this context, various issues that may be difficult to handle in the general case, for example identification of the largest two-sided quasigroup image, become more tractable. An appropriate universal algebraic language for affine diquasigroups is established, and the entropic models of this language are characterized. Various interesting classes of linear and affine diquasigroups are singled out for special attention, such as internally associative, Bol, and symmetric diquasigroups. The problem of determining which linear diquasigroups have an abelian group as their undirected replica is raised. One sufficient condition is provided, formulated in terms of a differential calculus for one-sided quasigroup words.

In this paper, we establish the transfer of the notion of Henriksen rings to direct products, trivial ring extensions, pullbacks and the amalgamation of rings. Our result generate new class of non-adequate Henriksen rings.

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be a Hall $s$-semiembedded subgroup of $G$ if $H$ is a Hall subgroup of $〈H,P〉$ for any $P\in {\mathrm{\text{Syl}}}_p(G)$, where $(p,|H|)=1$. In this paper, we investigate the influence of Hall $s$-semiembedded subgroups on the structure of the finite group $G$. Some new results about $G$ to be a $𝔉$-group are obtained, where $𝔉$ is a saturated formation.

By a long standing conjecture of Rédei if an elementary $p$-group of rank $3$ is a direct product of two of its normalized subsets, then at least one of the factors is contained in a subgroup of order $p^2$. Motivated by Rédei’s conjecture we decompose an elementary $2$-group into a direct product of more than two factors such that each factor spans the whole group.