Elementary properties of linear and algebraic groups
The so-called elementary properties of an algebraic system \(G\) are properties of \(G\) that can be expressed in the language of the first-order predicate calculus, where the basic operations are the operations of the system \(G\). Two algebraic systems \(G_1\) and \(G_2\) are said to have the same arithmetical type (or they are elementarily equivalent) if they have the same elementary properties. Elementary properties of linear groups were first studied by \textit{A. I. Mal'tsev} [Usp. Mat. Nauk 16, No. 3, 3-60 (1961; Zbl 0129.25903)]. Let \(G\) be the group GL, PG, SL, or PSL. Let \(m\geq n\geq 3\) be integers, and \(K_1\) and \(K_2\) be fields. Mal'tsev proved that \(G(m,K_1)\) and \(G(n,K_2)\) have the same arithmetical type if and only if \(m=n\) and the fields \(K_1\) and \(K_2\) have the same arithmetical type. Later, the theory has been developed for the cases of general linear groups over prime associative rings and skewfields, unitary groups over fields, and also some other algebraic structures. This article reviews the progress of this theory. The last section of the article is devoted to the elementary properties of Chevalley groups over algebraically closed fields of characteristic different from two.
scientific article; zbMATH DE number 1777801
scientific article