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International Journal of Mathematics and Mathematical Sciences
Volume 4, Issue 2, Pages 279-287
http://dx.doi.org/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 Hindawi Publishing Corporation. 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 AB 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 m2, 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.