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Classical groups over rings. (English) Zbl 0577.20033
Quadratic and Hermitian forms, Conf. Hamilton/Ont. 1983, CMS Conf. Proc. 4, 131-140 (1984).
[For the entire collection see Zbl 0542.00004.] The paper is a survey on the subject mentioned in the title. For a division ring A, the definitions of classical groups over A are given. In particular, for any $n\times n$ matrix Q over A, its ”orthogonal” group $\{\alpha \in GL\sb nA:$ $\alpha\sp*Q\alpha -Q$ is an alternating matrix$\}$ is defined. Normal subgroups, presentations and isomorphisms of classical groups over rings are discussed. Some open problems are formulated. One of these problems is the following one. Let A be a non- commutative division ring, q a non-singular pseudo-quadric form of Witt index at least 1. Then the group [GO(q),GO(q)]/center is simple. The author writes: ”This is known in many cases (but I could not prove this in some cases even for A finite dimensional over its center). Also, [O(q),O(q)] is simple in all cases.”
Reviewer: G.A.Margulis

MSC:
20G35Linear algebraic groups over adèles and other rings and schemes
20F05Generators, relations, and presentations of groups
20E36Automorphisms of infinite groups
20E07Subgroup theorems; subgroup growth