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Groups with triality, which arose in the papers of Glauberman and Doro, are naturally connected with Moufang loops. In this paper, we describe all possible, in a sense, groups with triality associated with a given Moufang loop. We also introduce several universal groups with triality and discuss their properties. [ABSTRACT FROM AUTHOR]

In this paper, we reexamine the foundation of Isaacs' π-theory. One of the key concepts in Isaacs' π-theory is the construction of the characters Bπ(G) for a π-separable group G. The key to determining which characters lie in Bπ(G) was the construction of a nucleus for each irreducible character χ. In this paper, we present a different way of finding a nucleus for χ which is based on a chain of normal subgroup $\mathcal{N}$. Using this nucleus, we obtain the set of characters $B_{\pi}(G:\mathcal{N})$. We investigate the properties that $B_{\pi}(G:\mathcal{N})$ has in common with Bπ(G). [ABSTRACT FROM AUTHOR]

(Co)induction functors appear in several areas of Algebra in different forms. Interesting examples are the so called induction functors in the Theory of Affine Algebraic Groups. In this paper we investigate Hopf pairings (bialgebra pairings) and use them to study (co)induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of Hopf pairings (bialgebra pairings) satisfying the so called α-condition. For those pairings the coinduction functor is studied and nice descriptions of it are obtained. Along the paper several interesting results are generalized from the case of base fields to the case of arbitrary commutative (Noetherian) ground rings. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a finite group G is isomorphic to the alternating group An with n≥5 if and only if they have the same set of the orders of solvable subgroups. [ABSTRACT FROM AUTHOR]

Let p and q be odd primes, and let G be an M-group of order paqb. Loukaki has proven that if N is a normal subgroup of G, then N is an M-group. In this paper, we present a simpler proof of Loukaki's theorem. We also present a slight generalization of Loukaki's theorem. [ABSTRACT FROM AUTHOR]

In this paper branching rules for the polynomial irreducible representations of the general linear groups in positive characteristic with highest weights labeled by partitions of the form (2a, 1b, 0c) and their restrictions to the special linear groups are found. The submodule structure of the restrictions of the corresponding irreducible modules for the group GLn(F) (or SLn(F)) to the naturally embedded subgroup GLn-1(F) (or SLn-1(F)) is determined. As a corollary, inductive systems of irreducible representations for GL∞(F) and SL∞(F) that consist of representations indicated above, are classified. The submodule structure of the relevant Weyl modules is refined. [ABSTRACT FROM AUTHOR]

A binomial f belonging to a toric ideal I is indispensable if, for any system $\mathcal{F}$ of binomial generators of I, either f or -f belongs to $\mathcal{F}$. In the present paper, we study indispensable binomials of the toric ideals IG arising from a finite graph G. First, we show that the toric ideal IG arising from a finite graph G whose complementary graph is weakly chordal is generated by the indispensable binomials if and only if no complete graph of order ≥4 is a subgraph of G. Second, we completely classify indispensable binomials of the toric ideal IG arising from a finite graph G satisfying the odd cycle condition. Finally, the existence of indispensable binomials of IG will be discussed. [ABSTRACT FROM AUTHOR]

A subnormal subgroup X of G has the following property: there exists a Dirichlet polynomial Q(s) with integer coefficients such that, for each t ∈ ℕ, Q(t) is the conditional probability that t random elements generate G given that they generate G together with the elements of X In this paper we analyze how far can a subgroup X be with this property from being a subnormal subgroup. [ABSTRACT FROM AUTHOR]

Given a compactly generated triangulated category $\mathcal{T}$ and a generating set ℛ of compact objects, the class of ℛ-injective objects $\mathsf{Inj}_\mathcal{R}\ \mathcal{T}$ is introduced. If $X\in\mathcal{T}$ is compact and ℛ={ΣnX}n∈ℤ it is shown that there is a functor \[ \Gamma:\mathsf{Inj}\ \mathcal{S} \to \mathsf{Inj}_{\mathcal R}\ \mathcal{T} \] identifying the class $\mathsf{Inj}\,\mathcal{S}$ of injective modules over the ℤ-graded ring $S=\bigoplus_{n\in\mathbb{Z}}\ \mathcal{T}(\Sigma^nX,X)$ with the class $\mathsf{Inj}_{\mathcal R}\ \mathcal{T}$. Also, the Ziegler and Zariski spectra of $\mathcal{T}$ and of S are discussed in the paper. [ABSTRACT FROM AUTHOR]

A simple direct method of constructing a faithful, irreducible representation of Sn of degree n - 1 is given. [ABSTRACT FROM AUTHOR]

We study the composition of F. R. Cohen's map Pn → Pnk with the Gassner representation, where Pn is the pure braid group. This gives us a linear representation of Pn whose composition factors are one copy of the Gassner representation of Pn and k - 1 copies of a diagonal representation, hence a direct sum of one-dimensional representations. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic p ≠ 2,3 and let G be a finite group. Necessary and sufficient conditions for δ3(U(KG)) = 1, where U(KG) is the unit group of the group algebra KG, are obtained. [ABSTRACT FROM AUTHOR]

G. Navarro [10] proposed a refinement of the McKay conjecture involving a special class of automorphisms. We verify that this new conjecture holds in the case of the alternating groups for p = 2. [ABSTRACT FROM AUTHOR]

Let G be a finite group and X a conjugacy class of G. We denote rank(G:X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks of the Conway groups Co2 and Co3. Computations were carried with the aid of computer algebra system 픾픸ℙ [18]. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring, G a group and RG its group ring. Let φ: RG → RG denote the R-linear extension of an involution φ defined on G. An element x in RG is said to be antisymmetric if φ(x) = -x. A characterization is given of when the antisymmetric elements $(RG)^{-}_{\varphi}$ of RG commute except when Char(R) = 3. [ABSTRACT FROM AUTHOR]

We calculate the invariant subspaces in the linear representation of the group of algebraic automorphisms of ℂn on the vector space of algebraic vector fields on ℂn and more generally we do this in a setting with parameter. As an application to the field of Several Complex Variables we get a new proof of the Andersén–Lempert observation and a parametric version of the Andersén–Lempert theorem. Further applications to the question of embeddings of ℂk into ℂn are announced. [ABSTRACT FROM AUTHOR]

Let (H, A) be a primitive central extension of a finite group G. We show that A is not characteristic in H in general, and further demonstrate in a series of examples that results, which hold about inner automorphisms of H, do not extend to the full automorphism group of H. We also give some new results about isoclinism and representation groups of G in the case that H is capable. Finally we give an example of two non-isomorphic groups of the same order, which not only have a representation group in common, but also have identical projective character tables. [ABSTRACT FROM AUTHOR]

We introduce the notion of almost finite dimensional representability of algebras and study its connection with the classical finiteness conditions. [ABSTRACT FROM AUTHOR]

A modular lattice L with 0 and 1 is called quotient finite dimensional (QFD) if [x,1] has no infinite independent set for any x∈L. We extend some results about QFD modules to upper continuous modular lattices by using Lemonnier's Lemma. One result says that QFD for a compactly generated lattice L is equivalent to Condition (C): for every m∈L, there exists a compact element t of L such that t∈[0,m] and [t,m[ has no maximal element. If L is not compactly generated, then QFD and (C) separate into two distinct conditions, which are analyzed and characterized for upper continuous modular lattices. We also extend to upper continuous modular lattices some characterizations of QFD modules with Gabriel dimension. Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory. [ABSTRACT FROM AUTHOR]

Let Ω be a finite linearly ordered set and let k be a positive integer. A permutation group G on Ω is called co-k-restricted min-wise independent on Ω if \[ \Pr(\min \pi(X)=\pi (x))=\frac{1}{|X|} \] for any X⊆Ω such that |X|≥|Ω|-k+1 and for any x∈X. We show that co-k-restricted min-wise independent groups are exactly the groups with the property that for each subset X⊆Ω with |X|≤k-1, the stabilizer G{X} of X in G is transitive on Ω\X. Using this fact, we determine all co-k-restricted min-wise independent groups. [ABSTRACT FROM AUTHOR]