International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 2, Pages 279-287
doi:10.1155/S0161171281000161
A note of equivalence classes of matrices over a finite field
1Department of Mathematical Sciences, Clemson University, Clemson, South Carolina, USA
2Department of Mathematics, The Pennsylvania State University, Sharon, Pennsylvania, USA
Received 16 July 1980
Copyright © 1981 J. V. Brawley and 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.
Abstract
Let Fqm×m denote the algebra of m×m matrices over the finite field Fq of q elements, and let Ω denote a group of permutations of Fq. It is well known that each ϕϵΩ can be represented uniquely by a polynomial ϕ(x)ϵFq[x] of degree less than q; thus, the group Ω naturally determines a relation ∼ on Fqm×m as follows: if A,BϵFqm×m then A∼B if ϕ(A)=B for some ϕϵΩ. Here ϕ(A) is to be interpreted as substitution into the unique polynomial of degree <q which represents ϕ.
In an earlier paper by the second author [1], it is assumed that the relation ∼ is an equivalence relation and, based on this assumption, various properties of the relation are derived. However, if m≥2, the relation ∼ is not an equivalence relation on Fqm×m. It is the purpose of this paper to point out the above erroneous assumption, and to discuss two ways in which hypotheses of the earlier paper can be modified so that the results derived there are valid.