Algebra Colloquium

A limit variety is a variety that is minimal with respect to being non-finitely based. Since the turn of the millennium, much attention has been given to the classification of limit varieties of aperiodic monoids. Seven explicit examples have so far been found, and the task of locating other examples has recently been reduced to two subproblems, one of which is concerned with monoids that satisfy the identity $xsxt\approx xsxtx$. We provide a complete solution to this subproblem by showing that there are precisely two limit varieties that satisfy this identity. One of them turns out to be the first example having infinitely many subvarieties. It is also deduced that the variety generated by any monoid of order 5 or less contains at most countably many subvarieties.

Let $\mathcal{S}$ be a Serre class of the category of modules and $\mathfrak{a}$ an ideal of a commutative noetherian ring $R$. Generalizations of depth and cohomological dimension with respect to $\mathcal{S}$for complexes are given. Some characterizations of these notions in terms of RHom, Koszul cohomology and local cohomology are found. The membership of the local cohomology ${\mathrm{H}}_{\mathfrak{a}}^i(X)$ of an $R$-complex $X$ in $\mathcal{S}$ is studied from below ($i<n$) and from above ($i>n$), which in special cases recovers and generalizes the known results about the usual depth and cohomological dimension of modules.

We give the explicit formula for the characteristic polynomial of the Laplacian matrix of every rooted balanced $k$-ary tree of depth $r$.

Let a finite group $A$ act on a finite group $G$ via automorphism and let $s$ be a positive integer. In this paper, we introduce the probabilistic zeta function of ordered $s$-tuples from $G$ which $A$-generate $G$, denoted by ${\mathrm{P}}^A(G,s)$. When $(|A|,|G|)=1$, it is proved that $G$ is solvable if and only if ${\mathrm{P}}^A(G,s)$ has an Euler product expansion with all factors of the form $1-c/q^s$, where each $q$is a prime power. This is a generalization of Detomi and Lucchini's result in 2003.

Let $p$ be a prime and $\mathcal{F}$ be a fusion system over a finite $p$-group $S$. The fusion system $\mathcal{F}$ is said to be nilpotent if $\mathcal{F}={\mathcal{F}}_S(S)$. Using $\mathcal{F}$-strongly closed subgroups, we provide a new criterion for a saturated fusion system $\mathcal{F}$ to be nilpotent, which may be viewed as extending the Glauberman-Thompson $p$-nilpotency criterion to fusion systems.

For an algebraically closed field $\mathbb{F}$ of characteristic different from 2 and a field $\mathbb{K}$ satisfying that for all $a$, $b\in \mathbb{K}$, $a^2+b^2=0$ if and only if $a=b=0$, we extend the notion of superderivations to special graded local superderivations, graded local superderivations and local superderivations. Applying matrix theory, we establish a calculation method for local superderivations. Then we determine all the special graded local superderivations of Heisenberg superalgebras of even center over $\mathbb{F}$and graded local superderivations and local superderivations of Heisenberg superalgebras of even center over $\mathbb{K}$. Finally, we determine all the special graded local superderivations, graded local superderivations and local superderivations of Heisenberg superalgebras of odd center over $\mathbb{F}$.

In this article, we study the projective class ring of bosonization $B(G,\underset{\_}{g},\underset{\_}{\chi })$ of quantum linear space $\mathrm{\Re }$. We classify all finite-dimensional indecomposable projective $B(G,\underset{\_}{g},\underset{\_}{\chi })$-modules and simple $B(G,\underset{\_}{g},\underset{\_}{\chi })$-modules. The decomposition rules for tensor products of indecomposable projective $B(G,\underset{\_}{g},\underset{\_}{\chi })$-modules and simple $B(G,\underset{\_}{g},\underset{\_}{\chi })$-modules are given, and the structures of projective class ring $r_p(B(G,\underset{\_}{g},\underset{\_}{\chi }))$ are described. It turns out that $r_p(B(G,\underset{\_}{g},\underset{\_}{\chi }))$ is a commutative ring generated by $n+1$ elements subject to certain relations. Finally, we classify all finite-dimensional indecomposable modules over the complexified projective class ring$R_p(B(G,\underset{\_}{g},\underset{\_}{\chi }))$.

The intersection power graph of a finite group $G$ is a simple graph whose vertex set is $G$, in which two distinct vertices $x$ and $y$ are adjacent if and only if either one of $x$ and $y$ is the identity element, or $⟨x⟩\cap ⟨y⟩$ is non-trivial. A number of important graph classes, including cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. In this paper, we characterize the finite groups whose intersection power graphs are cographs, split graphs, and threshold graphs. We also classify the finite nilpotent groups whose intersection power graphs are chordal.

The enhanced power graph ${\mathcal{P}}_E(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x$, $y$ are adjacent if $x$, $y\in ⟨z⟩$ for some $z\in G$. We give an affirmative answer to the question posed by Cameron which asks whether the complement of the enhanced power graph ${\mathcal{P}}_E(G)$ of a non-cyclic group$G$, $\overline{{\mathcal{P}}_E(G)}$, has only one connected component apart from isolated vertices. We classify all finite groups $G$ for which $\overline{{\mathcal{P}}_E(G)}$ is bipartite. We show that the graph $\overline{{\mathcal{P}}_E(G)}$ is weakly perfect. Further, we study the subgraph $\overline{{\mathcal{P}}_E(G^{\ast })}$ of $\overline{{\mathcal{P}}_E(G)}$ induced by all the non-isolated vertices of$\overline{{\mathcal{P}}_E(G)}$. We classify all finite groups$G$ with the graph $\overline{{\mathcal{P}}_E(G^{\ast })}$ being unicyclic and pentacyclic, and we prove the non-existence of finite groups $G$ such that the graph $\overline{{\mathcal{P}}_E(G^{\ast })}$ is bicyclic, tricyclic or tetracyclic. Finally, we characterize all finite groups $G$ for which the graph $\overline{{\mathcal{P}}_E(G^{\ast })}$ is outerplanar, planar, projective-planar and toroidal separately.

Quotient canonical $(m,n)$-hypermodules over Krasner $(m,n)$-hyperrings are studied as a generalization of the well-known algebraic hyperstructures. In this work, we prove that if $N$ is a normal $(m,n)$-ary subhypermodule, then the $m$-ary hyperoperations defined on the quotient canonical $(m,n)$-ary hypermodule $[M:N^{\ast }]$ are $m$-ary operations and the relation $N^{\ast }$ is a strongly regular relation. Further, in this case, it is shown that the scalar hyperoperation is just an operation.

In this paper, we study quasitriangular structures on a class of semisimple Hopf algebras ${\mathbb{k}}^G{\mathrm{\#}}_{\sigma ,\tau }\mathbb{k}{\mathbb{Z}}_2$ constructed through abelian extensions of $\mathbb{k}{\mathbb{Z}}_2$ by ${\mathbb{k}}^G$ for an abelian group $G$. We find that there is an analogy between these quasitriangular structures and the solutions of a linear system.

In this paper, we discuss 3-transposition groups. In particular, we find sizes of maximal symmetric subgroups of these groups, which are in Fischer's list. In addition, we build faithful representations of symmetric groups in orthogonal, symplectic, and unitary spaces, which fully verify the main result. Thus, we find candidate groups for which one may be able to build standard Majorana representations.

We provide a detailed description of inner and outer Poisson structures for the trivial extension $A⋉D(A)$, where $A$ is a hereditary algebra of Dynkin type ${\mathbb{A}}_n$ and $D(A)={\mathrm{Hom}}_K(A,K)$.