Some groups generated by transvection subgroups (Q1176662)

Let \(k\) be a field (not necessarily commutative), \(V\) a (not necessarily finite-dimensional) \(k\)-vector space and \(V^*\) the dual of \(V\). If \(\varphi\) and \(x\) are nonzero elements in \(V^*\) and \(V\), respectively, then \(t(\varphi,x): V\to V: v\to v+v\varphi x\) is a \(k\)-transvection and \(T(\varphi,x)=\langle t(\varphi',x')\mid\) \(\varphi'\in\langle\varphi\rangle\), \(x'\in \langle x\rangle\rangle\) a \(k\)-transvection subgroup. Under certain assumptions the authors describe groups \(G\) of linear transformations generated by \(k\)-transvection subgroups. The infinite-dimensional situation is thoroughly discussed and illustrated by a number of examples. The special case where \(V\) is finite-dimensional and \(G\) is irreducible is due to J. E. McLaughlin.