# Soluble Linear Groups

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## Abstract

Let
are isomorphic as groups, both being infinite cyclic, but have very different properties as linear groups. If

*F*be a field and n a positive integer.*GL*(*n*,*F*) denotes the multiplicative group of all n by*n*invertible matrices with entries in*F*. By definition, a*linear*group is a subgroup of*GL*(*n*,*F*) for some*n*and*F*. Warning: a linear group is more that just a group; it is a group together with a particular embedding into some selected*GL*(*n*,*F*). For example, working over the complex numbers the groups$$X=\left\langle \left(\begin{array}{@{}c@{\quad}c@{}}1&0\\1&1\end{array}\right)\right\rangle \quad \mbox{and}\quad Y=\left\langle \left(\begin{array}{@{}c@{\quad}c@{}}2&0\\0&1\end{array}\right)\right\rangle $$

*R*is a ring (with an identity as always), then*GL*(*n*,*R*) denotes the obvious thing, namely the group of*n*by*n*invertible matrices over the ring*R*, but its subgroups will not be called linear groups unless, of course,*R*is a (commutative) integral domain.## Keywords

Normal Subgroup Linear Group Inverse Image Matrix Ring Zariski Topology
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag London 2009