Related papers: Locally isotropic elementary groups
We give a description of elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that elementary subgroups of finitely generated free groups are free factors.…
In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of…
In this paper, we prove that the endomorphism rings End A and End A' of periodic infinite Abelian groups A and A' are elementarily equivalent if and only if the endomorphism rings of their p-components are elementarily equivalent for all…
Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic p, extending the construction of Deligne and Lusztig of representations of reductive groups over…
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…
In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank >1 over a commutative ring (with 1/2 for the systems A_2, F_4, B_l, C_l; with 1/2 and 1/3 for the system G_2) is…
We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine…
We identify the class of elementary groups: the smallest class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contains the profinite groups and the discrete groups, is closed under group extensions of…
Finite non-abelian non-metacyclic $2$-generated $p$-groups (${p>2}$) of nilpotency class $2$ with cyclic commutator subgroup which are the additive groups of local nearrings are described. It is shown that the subgroup of all non-invertible…
In this paper we provide a complete algebraic characterization of elementary equivalence of rings with a finitely generated additive group in the language of pure rings. The rings considered are arbitrary otherwise.
We give a simplified proof of Tits' classification of semisimple algebraic groups that remains valid over semilocal rings. In particular, we provide explicit necessary and sufficient conditions that anisotropic groups of a given type appear…
Let $\mathcal{O}_2$ and $\mathcal{O}'_2$ be two distinct finite local rings of length two with residue field of characteristic $p$. Let $\mathbb{G}(\mathcal{O}_2)$ and $\mathbb{G}(\mathcal{O}'_2)$, be the group of points of any reductive…
In this paper, we continue our study of abstract representations of elementary subgroups of Chevalley groups of rank $\geq 2.$ First, we extend our earlier methods to analyze representations of elementary groups over arbitrary associative…
For an isotropic reductive group G satisfying a suitable rank condition over an infinite field k, we show that the sections of the $\mathbb{A}^1$-fundamental group sheaf of G over an extension field L/k can be identified with the second…
Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity,…
The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is fully elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The…
The main result of this paper is the following theorem. Let q be a prime, A an elementary abelian group of order q^3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that C_G(a)' is periodic…
The present work investigates regular, semiregular, and chiral polytopes of any rank $d\geq 3$, whose automorphism groups are 2-groups. There is a large variety of rather small finite regular or alternating semiregular polytopes with…
Let $R$ be an associative ring with 1, $G=GL(n, R)$ be the general linear group of degree $n\ge 3$ over $R$. In this paper we calculate the relative centralisers of the relative elementary subgroups or the principal congruence subgroups,…
We show that the standard set of elementary generators of an elementary isotropic reductive group over a connected finitely generated ring is a Kazhdan subset. This generalizes the corresponding result of M. Ershov, A. Jaikin-Zapirain, and…