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We show that semi-infinite cohomology of a finite dimensional graded algebra (satisfying some additional requirements) are a particular case of a general categorical construction. The motivating example is provided by small quantum groups…

We prove for residually finite groups the following long standing conjecture: the number of twisted conjugacy classes of an automorphism of a finitely generated group is equal (if it is finite) to the number of finite dimensional…

We extend to the context of algebraic groups a classic result on extensions of abstract groups relating the set of isomorphism classes of extensions of $G$ by $H$ with that of extensions of $G$ by the center $Z$ of $H$. The proof should be…

In this paper we first survey some basic results in the cohomology of finite groups, and then discuss recent work on constructing free actions of finite groups on products of spheres.

We prove continuity results for abstract epimorphisms of locally compact groups onto finitely generated groups.

We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds…

It is the goal of this article to extend the notion of quantization from the standard interpretation focused on non-commuting observables defined starting from classical analogues, to the topological equivalents defined in terms of…

In this paper, we discuss some well-known results and some open problems of the theory of $\sigma$-properties of a group related to the study of generalized $T$-groups.

In the paper new criteria of existence and conjugacy of Hall subgroups of finite groups are given.

This paper is an attempt to find out which properties of a finite group G can be expressed in terms of commutators of elements of coprime orders. A criterion of solubility of G in terms of such commutators is obtained. We also conjecture…

We complete the description of group gradings on finite-dimensional incidence algebras. Moreover, we classify the finite-dimensional graded algebras that can be realized as incidence algebras endowed with a group grading.

We perform the computations necessary to establish a multiplicity one statement for the irreducible representations of a finite spin group which in turn yields the classification of irreducible representations of finite spin groups. (The…

In this short note, we introduce a generalization of the canonical base property, called transfer of internality on quotients. A structural study of groups definable in theories with this property yields as a consequence infinitely many new…

The present survey aims at being a list of Conjectures and Problems in an area of model-theoretic algebra wide open for research, not a list of known results. To keep the text compact, it focuses on structures of finite Morley rank,…

This paper reproduces the text of a part of the Author's DPhil thesis. It gives a proof of the classification of non-trivial, finite homogeneous geometries of sufficiently high dimension which does not depend on the classification of the…

We study finiteness (and vanishing) properties of the higher order degrees associated to complements of complex affine plane curves with mild singularities at infinity. Our results impose new obstructions on the class of groups that can be…

In the paper it is proven that Carter subgroups of a finite group are conjugate. A complete classification of Carter subgroups in finite almost simple groups is also obtained.

We give new upper and lower bounds on the concavity of quantum entropy. Comparisons are given with other results in the literature.

We report on recent progress concerning the relationship that exists between the algebraic structure of a finite group and certain features of its class-size prime graph.

We introduce the new notion of quotient-saturation as a measure of the immensity of the quotient structure of a group. We present a sufficient condition for a finitely presented group to be quotient-saturated, and use it to deduce that…