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.