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In this paper, the derivation algebra and the automorphism group of the twisted N=2 superconformal algebra are completely determined. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new example of Hopf group coalgebras by considering Radford's biproduct Hopf algebra in the Turaev category. Furthermore, we find some sufficient and necessary conditions for such Radford's biproduct Hopf algebra to admit quasitriangular structures in the sense of Turaev group coalgebras. [ABSTRACT FROM AUTHOR]

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ∗-clean, where R is a commutative local ring and G is one of the groups C3, C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ∗-clean. [ABSTRACT FROM AUTHOR]

Let β be a Lie algebra of Block type with basis {Lɑ, i | ɑ ... Z, i ... Z+} and relations [Lɑ, i, Lβ, j] = (ɑ -- 1)(j + 1) -- (β -- l)(i + l))Lɑ+β,i+j. In the present paper, the derivation algebra and automorphism group of B are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of β is trivial. [ABSTRACT FROM AUTHOR]

Let G be a finite classical group of characteristic p. In this paper, we give an arithmetic criterion of the primes r ≠ p, for which the Steinberg character lies in the principal r-block of G. The arithmetic criterion is obtained from some combinatorial objects (the so-called partition and symbol). [ABSTRACT FROM AUTHOR]

In this paper, we classify and determine the structure of finite groups whose character degree graphs are empty graphs [ABSTRACT FROM AUTHOR]

In this paper we continue to study the lifting of outer actions of groups and introduce the concept of equivalence of two lifting homomorphisms, maximal lifting homomorphisms and maximal split extensions. Some criteria are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of n-isoclinism to the class of all pairs of groups (G,M), where M is a normal subgroup of a group G. The details of that notion are studied and some equivalent conditions are given for two pairs of groups to be n-isoclinic. In addition, the concept of subgroup- and quotient-irreducibility for a pair of groups is defined and we derive how to see whether (G,M) has a pair of subgroups or a pair of factor groups n-isoclinic with (G,M). [ABSTRACT FROM AUTHOR]

Let χ be a character of order N=2t(t ⩾ 4) over a finite field 픽q = 픽pf with p ≡ 7, 9 (mod 16). In this paper, we evaluate explicitly the Gauss sums G(χr) related to the group generated by the character χ. [ABSTRACT FROM AUTHOR]

We take in this paper an arbitrary class $\mathcal{K}$ of groups as a base, and define a radical property 풫 for which every group in $\mathcal{K}$ is 풫-semisimple. This is called the upper radical property determined by the class $\mathcal{K}$. At the same time, we define a radical property 풫 for which every group in $\mathcal{K}$ is a 풫-radical group. This is called the first lower radical property determined by the class $\mathcal{K}$. Also, we give another construction leading to the second lower radical property which is proved to be identical with the first one. [ABSTRACT FROM AUTHOR]

In this paper, Lie bialgebra structures on the extended Schrödinger-Virasoro Lie algebra $\mathcal{L}$ are classified. It is obtained that all the Lie bialgebra structures on $\mathcal{L}$ are triangular coboundary. As a by-product, it is derived that the first cohomology group $H^1(\mathcal{L},\mathcal{L}\otimes\mathcal{L})$ is trivial. [ABSTRACT FROM AUTHOR]

Let G be a group and let w = w(x1, x2,..., xn) be a word in the absolutely free group Fn on free variables x1, x2,..., xn. The set S(n)(G) of all words w such that the equality w(gσ1, gσ2,..., gσn) = w(g1, g2,..., gn) holds for all g1, g2,..., gn∈G and all permutations σ ∈ Sn is a subgroup of Fn, called the subgroup of n-symmetric words for G. In this paper, the groups S(2)(Dp) and S(3)(Dp) for dihedral groups Dp are determined, where p > 3 is a prime. In particular, it turns out that the groups S(3)(Dp) are not abelian. [ABSTRACT FROM AUTHOR]

Let G be a finite group and p a prime. We say that G is quasisupersoluble (resp., p-quasisupersoluble) if for every non-cyclic chief factor H/K of G (resp., for every non-cyclic chief factor H/K of G of order divisible by p), every automorphism of H/K induced by an element of G is inner. In this paper, we study the structure of quasisupersoluble and p-quasisupersoluble finite groups. [ABSTRACT FROM AUTHOR]

Let G be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with $\omega(\mathcal{A}_G)\leq 57$, where 57 is the clique number of the non-commuting graph of the projective special linear group PSL(2,7). We also determine $\omega(\mathcal{A}_G)$ for all finite minimal simple groups G. [ABSTRACT FROM AUTHOR]

In this paper, we define a new type of configurations as two-sided configurations, and investigate which group properties can be characterized by them. It is proved that for polycyclic torsion free groups, having the same finite quotient sets does not imply the (two-sided) configuration equivalence. We show that isomorphisms and configuration equivalences coincide for some free products of groups and a class of nilpotent groups. [ABSTRACT FROM AUTHOR]

Given two numerical semigroups S and S', the distance between S and S' is the cardinality of S\S' plus the cardinality of S'\S. In this paper we study those numerical semigroups S for which there is a symmetric numerical semigroup whose distance to S is one. [ABSTRACT FROM AUTHOR]

Let b(G) denote the largest irreducible character degree of a finite group G. In this paper, we prove that if G is a solvable group which does not involve a section isomorphic to the wreath product of two groups of equal prime order p, and if b(G) < pn, then |G:Op(G)|p < pn. [ABSTRACT FROM AUTHOR]