Soluble Linear Groups

  • B. A. F. WehrfritzEmail author
Part of the Algebra and Applications book series (AA, volume 10)


Let 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
$$X=\left\langle \left(\begin{array}{@{}c@{\quad}c@{}}1&0\\1&1\end{array}\right)\right\rangle \quad \mbox{and}\quad Y=\left\langle \left(\begin{array}{@{}c@{\quad}c@{}}2&0\\0&1\end{array}\right)\right\rangle $$
are isomorphic as groups, both being infinite cyclic, but have very different properties as linear groups. If 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.


Normal Subgroup Linear Group Inverse Image Matrix Ring Zariski Topology 
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Copyright information

© Springer-Verlag London 2009

Authors and Affiliations

  1. 1.School of Mathematical SciencesQueen Mary, University of LondonLondonUK

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