International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 3, Pages 503-512

Permutation matrices and matrix equivalence over a finite field

Gary L. Mullen

Department of Mathematics, The Pennsylvania State University, Sharon 16146, Pennsylvania, USA

Received 21 March 1980; Revised 12 August 1980

Copyright © 1981 Gary L. Mullen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let F=GF(q) denote the finite field of order q and Fm×n the ring of m×n matrices over F. Let 𝒫n be the set of all permutation matrices of order n over F so that 𝒫n is ismorphic to Sn. If Ω is a subgroup of 𝒫n and A, BϵFm×n then A is equivalent to B relative to Ω if there exists Pϵ𝒫n such that AP=B. In sections 3 and 4, if Ω=𝒫n formulas are given for the number of equivalence classes of a given order and for the total number of classes. In sections 5 and 6 we study two generalizations of the above definition.