Decomposition of transvections for automorphisms. (Q844485)

Let \(R\) be a commutative ring and \(n\geq 4\). It is known that all automorphisms of \(\text{GL}(n,R)\) are `standard'. Let \(P_i\) denote the maximal parabolic subgroup consisting of the \(g\) that have zeroes in the last \(n-i\) rows of the submatrix formed by the first \(i\) columns. We say that a parabolic subgroup is of type \(i\) if it is conjugate to \(P_i\) under a permutation matrix. Let \(\varphi\) be an automorphism of \(\text{GL}(n,R)\). The last theorem in the paper states that the elementary subgroup \(E(n,R)\) is generated by transvections \(g\) such that \(g\) itself lies in a parabolic of type \(1\), whereas \(\varphi(g)\) lies in the intersection of two parabolics, one of type \(1\) the other of type \(n-1\). This theorem easily implies that \(\varphi\) is `standard', but that was already used in the proof. So it is desirable to find a different argument. The paper contains several similar results and discusses the connection with some basic problems about \(\text{GL}(n,R)\) and \(E(n,R)\).