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We introduce and develop the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary…
Free groups are known to be homogeneous, meaning that finite tuples of elements which satisfy the same first-order properties are in the same orbit under the action of the automorphism group. We show that virtually free groups have a…
We consider a notion of "numerosity" for sets of tuples of natural numbers, that satisfies the five common notions of Euclid's Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a notion, we show…
We introduce a new class of locally compact groups, namely the strongly compactly covered groups, which are the Hausdorff topological groups $G$ such that every element of $G$ is contained in a compact open normal subgroup of $G$. For…
In this article we study the first, the second and the third homology groups of the elementary group $\textrm{E}_2(A)$, where $A$ is a commutative ring. In particular, we prove a refined Bloch-Wigner type exact sequence over a semilocal…
Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems…
Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results…
We develop a theory of descent and forms of tensor categories over arbitrary fields. We describe the general scheme of classification of such forms using algebraic and homotopical language, and give examples of explicit classification of…
We consider standard graded toric rings $R_{\Delta}$ whose generators correspond to the faces of a simplicial complex $\Delta$. When $R_{\Delta}$ is normal, it is shown that its divisor class group is free. For a flag complex $\Delta$ which…
In this article we provide a complete characterization of abelian group rings which are K\"{o}the rings. We also provide characterizations of (possibly non-abelian) group rings over division rings which are K\"{o}the rings, both in…
Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbert series, which is…
In a paper on the taxonomy of 2-primal rings, examples of various types of rings that are related to commutativity such as reduced, symmetric, duo, reversible and PS~I were given in order to show that the ring class inclusions were strict.…
In this paper, we introduce and study two new classes of commutative rings, namely semi transitional rings and transitional rings, which extend several classical ideas arising from rings of continuous functions and their variants. A general…
Schur rings are a type of subring of the group ring that is spanned by a partition of the group that meets certain conditions. Past literature has exclusively focused on the finite group case. This paper extends many classic results about…
We develop the theory of groupoid graded semisimple rings. Our rings are neither unital nor one-sided artinian. Instead, they exhibit a strong version of having local units and being locally artinian, and we call them $\Gamma_0$-artinian.…
Various descending chains of subgroups of a finite permutation group can be used to define a sequence of `basic' permutation groups that are analogues of composition factors for abstract finite groups. Primitive groups have been the…
In this paper we investigate a family of Hamiltonian-minimal Lagrangian submanifolds in ${\mathbb C}^m$, ${\mathbb C}P^m$ and other symplectic toric manifolds constructed from intersections of real quadrics. In particular, we explain the…
Every simple Hermitian Lie group has a unique family of spherical representations induced from a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. For most Hermitian groups, this family contains a complementary…
We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical…
We consider three families of groups: the Bianchi groups SL(2,O) where O is the ring of integers of an imaginary, quadratic field; the groups SL*(2,O) where O is a *-order of a definite, rational quaternion algebra with an orthogonal…