$GL_n(\mathbb{F}_q)$-analogues of some properties of $n$-cycles in $\mathfrak{S}_n$

Abstract

We give analogues in the finite general linear group of two elementary results concerning long cycles and transpositions in the symmetric group: first, that the long cycles are precisely the elements whose minimum-length factorizations into transpositions yield a generating set, and second, that a long cycle together with an appropriate transposition generates the whole symmetric group.