Algebra Colloquium

We prove that generalized free products of certain abelian subgroup separable groups are abelian subgroup separable. Applying this, we show that tree products of polycyclic-by-finite groups, amalgamating central subgroups or retracts are abelian subgroup separable.

Let G be a finite group and H a subgroup of G. We say that H is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a generalized smooth group (GS-group) if [G/L] is totally smooth for every subgroup L of G of prime order. In this paper, we investigate the structure of G under the assumption that each subgroup of prime order is S-permutable if the maximal subgroups of G are GS-groups.

We determine the Gerstenhaber algebra structure on the Hochschild cohomology ring of Temperley–Lieb algebras in this paper.

In this paper, we introduce Gorenstein weak injective and weak flat modules in terms of, respectively, weak injective and weak flat modules; the classes of Gorenstein weak injective and weak flat modules are larger than the classical classes of Gorenstein injective and flat modules. In this new setting, we characterize rings over which all modules are Gorenstein weak injective. Moreover, we discuss the relation between the weak cosyzygy and Gorenstein weak cosyzygy of a module, and also the stability of Gorenstein weak injective modules.

We study skew cyclic codes over a class of rings $R={\mathbb{F}}_0\oplus {\mathbb{F}}_1\oplus \cdots \oplus {\mathbb{F}}_{t-1}$, where each ${\mathbb{F}}_i$ $(i=0,\dots ,t-1)$ is a finite field. We prove that a skew cyclic code of arbitrary length over R is equivalent to either a usual cyclic code or a quasi-cyclic code over R. Moreover, we discuss possible extension of our results in the more general setting of ${\delta }_R$-dual skew constacyclic codes over R, where δ_R is an automorphism of R.

A group G is said to be (l, m, n)-generated if it is a quotient of the triangle group $T(l,m,n)=⟨x,y,z\mid x^l=y^m=z^n=xyz=1⟩$. Moori posed in 1993 the question of finding all the triples (l, m, n) such that non-abelian finite simple groups are (l, m, n)-generated. We partially answer this question for the Fischer sporadic simple group Fi₂₃. In particular, we investigate all (2, q, r)-generations for the Fischer sporadic simple group Fi₂₃, where q and r are distinct prime divisors of |Fi₂₃|.

The notion of DG-Gorenstein injective complexes is studied in this article. It is shown that a complex G is DG-Gorenstein injective if and only if G is exact with ${\mathrm{Z}}_n\left(G\right)$ Gorenstein injective in R-Mod for each $n\in \mathrm{\mathbb{Z}}$ and any morphism $f:E\to G$ is null homotopic whenever E is a DG-injective complex.

A semidirect product $G=F⋋H$ of groups F and H is called a Frobenius group if the following two conditions are satisfied: (F₁) H acts freely on F, that is, $f^h=f$ for f in F and h in H only if h = 1 or f = 1. (F₂) Every non-identity element $h\in H$ of finite order n induces in F by conjugation in G a splitting automorphism, that is, $f{f}^h\cdots f^{h^{n-1}}=1$ for every $f\in F$; in other words, the order of $f{h}^{-1}$ is equal to n. We describe the normal structure of a Frobenius group with periodic subgroup H generated by elements of order 3.

Let ${\mathbb{F}}_q$ be a finite field of any characteristic and $\mathrm{G}\mathrm{L}(n,{\mathbb{F}}_q)$ be the general linear group over ${\mathbb{F}}_q$. Suppose W denotes the standard representation of $\mathrm{G}\mathrm{L}(n,{\mathbb{F}}_q)$, and $\mathrm{G}\mathrm{L}(n,{\mathbb{F}}_q)$ acts diagonally on the direct sum of W and its dual space W^∗. Let G be any subgroup of $\mathrm{G}\mathrm{L}(n,{\mathbb{F}}_q)$. Suppose the invariant field ${\mathbb{F}}_q(W{)}^G={\mathbb{F}}_q(f_1,f_2,\dots ,f_k)$, where $f_1,f_2,\dots ,f_k$ in ${\mathbb{F}}_q[W{]}^G$ are homogeneous invariant polynomials. We prove that there exist homogeneous polynomials ${\ell }_1,{\ell }_2,\dots ,{\ell }_n$ in the invariant ring ${\mathbb{F}}_q[W\oplus W^{*}{]}^G$ such that the invariant field ${\mathbb{F}}_q(W\oplus W^{*}{)}^G$ is generated by $\{f_1,f_2,\dots ,f_k,{\ell }_1,{\ell }_2,\dots ,{\ell }_n\}$ over ${\mathbb{F}}_q$.

In this note we classify the compatible left-symmetric superalgebra structures over a super Virasoro type algebra, whose even part is the centerless twisted Heisenberg–Virasoro algebra, with some natural grading conditions.

We prove that if G is a gap-free and chair-free simple graph, then the regularity of the edge ideal of G is no more than 3. If G is a gap-free and P₄-free graph, then it is a chair-free graph; furthermore, the complement of G is chordal, and thus the regularity of G is 2.

Let H₃ be the 9-dimensional Taft Hopf algebra, let $r\left(H_3\right)$ be the corresponding Green ring of H₃, and let $\mathrm{Aut}\left(\mathrm{ℝ}\right(H_3\left)\right)$ be the automorphism group of Green algebra $\mathrm{ℝ}\left(H_3\right)=\mathrm{ℝ}{\otimes }_{\mathrm{\mathbb{Z}}}r\left(H_3\right)$ over the real number field ℝ. We prove that the quotient group $\mathrm{Aut}\left(\mathrm{ℝ}\right(H_3\left)\right)/T_1$ is isomorphic to the direct product of the dihedral group of order 12 and the cyclic group of order 2, where T₁ is the isomorphism class which contains the identity map and is isomorphic to a group $G=\left\{\right(c,d)\in {\mathrm{ℝ}}^2 | (c,d)\ne (-1/3,-1/6)\}$ with multiplication given by $(c_1,d_1)·(c_2,d_2)=(c_1+c_2+2c_1{c}_2-4d_1{d}_2+2c_1{d}_2+2d_1{c}_2, d_1+d_2-2c_1{c}_2-2d_1{d}_2+4c_1{d}_2+4d_1{c}_2)$.

For a group G and a non-empty subset Ω of G, the commuting graph $\mathcal{C}(G,\Omega )$ of Ω is a graph whose vertex set is Ω and any two vertices are adjacent if and only if they commute in G. Define $T_{4n}=⟨a,b\mid a^{2n}=b^4=1,a^n=b^2,b^{-1}ab=a^{-1}⟩$, the dicyclic group of order $4n$ $(n\ge 3)$, which is also known as the generalized quaternion group. We mainly investigate the properties and metric dimension of the commuting graphs on the dicyclic group $T_{4n}$.

In this paper, we consider the tensor product modules of a class of non-weight modules and highest weight modules over the Virasoro algebra. We determine the necessary and sufficient conditions for such modules to be simple and the isomorphism classes among all these modules. Finally, we prove that these simple non-weight modules are new if the highest weight module over the Virasoro algebra is non-trivial.

A ring is said to satisfy the strong 2-sum property if every element is a sum of two commuting units. In this note, we present some sufficient or necessary conditions for the matrix ring over a commutative local ring to have the strong 2-sum property.