On non-normal subgroups of \(GL_ n(A)\) which are normalized by elementary matrices (Q788115)

Let A be a ring and \({\mathfrak q}\) a two-sided ideal. Let \(\Delta\) (\({\mathfrak q})\) be the normal subgroup of \(GL_ n(A)\) generated by \({\mathfrak q}\)- elementary matrices, and \(H({\mathfrak q})\) the subgroup of matrices congruent modulo \({\mathfrak q}\) to xI, where x is central. If \({\mathfrak e}\) is the largest two-sided ideal with \(\Delta\) (\({\mathfrak e})\subset E\), then \(\Delta({\mathfrak e})\leq [G,E]\leq [G,H({\mathfrak e})].\) The paper investigated whether \([G,H({\mathfrak e})]\neq \Delta({\mathfrak e})\) implies the existence of a non-normal subgroup E. Examples show that the answer can be yes or no. General cases are given where the answer can be determined.