A Note of Equivalence Classes of Matrices over a Finite Field

Let [formula omitted] denote the algebra of mxm matrices over the finite field F_q of q elements, and let Ω denote a group of permutations of F_q. It is well known that each φεΩ can be represented uniquely by a polynomial φ(x)εF_q[x] of degree less than q; thus, the group Ω naturally determines a relation ~ on [formula omitted] as follows: if [formula omitted] then A~B if φ(Α) = B for some φεΩ. Here φ(Α) 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 [formula omitted]. 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.