# Generators of Automorphism Groups of Modules

• H. Ishibashi
Chapter
Part of the NATO ASI Series book series (ASIC, volume 333)

## Abstract

This note consists of two parts. In the first part we treat the length problem for unitary groups and symplectic groups over a quasisemilocal semihereditary ring. We treat orthogonal groups over valuation rings and quasisemilocal semihereditary domains. Further, we find some minimal or small sets of generators for these groups over finite fields. Throughout part one we assume that the rings have 2 as a unit and the modules on which these groups act are nonsingular.

In the second part, we treat the symplectic groups Sp(V) on a free module V of rank n over a local ring R with an alternating bilinear map $$f:V\times Vn \to R$$ Define $$M=\left\{{x\epsilon V|f(x,V)=R}\right\}$$ and for $$\theta \neq S\subseteq M$$ let TR(S) denote the subgroup generated by all transvections with axis s⊥ for s in S. Then T R(M) = Sp(V) if f is nonsingular, and we find necessary and sufficient conditions for S to satisfy TR(S) = T R(M). Further, we determine the smallest number of elements of a subset S’ of S such that T R(S′) = TR(M). Throughout part two, we do not assume that R has 2 as a unit, neither do we assume that V is nonsingular.

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