Powers of 1-cyclic conjugacy classes in the groups GL\(_ n(F)\) and SL\(_ n(F)\) (Q1355217)

The author defines an invertible \(n\times n\) matrix \(A\) over a field \(F\) to be 1-cyclic if it is similar to a block-direct sum of cyclic matrices, with at most one of the blocks of size \(1\times 1\). It is shown that if \(A\) is 1-cyclic with \(n\geq 2\) and \(|F|\geq 4\), then every nonscalar \(M\in\text{GL}_n(F)\) with \(\text{det }M=(\text{det }A)^4\) is a product of four matrices, each of which is similar to \(A\) by an element of \(\text{SL}_n(F)\). The result is applied to the problem of factoring elements of \(\text{SL}_n(F)\) into products of unipotent matrices of index two, and into products of matrices of a given finite order.