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

TL;DR

This paper explores properties of long cycles in the symmetric group and provides finite general linear group analogues, revealing new structural insights into these algebraic objects.

Contribution

It introduces finite group analogues of key symmetric group properties related to long cycles and transpositions, expanding understanding of linear groups over finite fields.

Findings

Long cycles in $GL_n(F_q)$ correspond to elements with minimal factorizations generating the group.

A long cycle combined with a suitable transposition generates the entire $GL_n(F_q)$.

Provides structural parallels between symmetric groups and finite linear groups.

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.