A note of equivalence classes of matrices over a finite field

Let F q m × m denote the algebra of m × m 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 ϕ ϵ Ω can be represented uniquely by a polynomial ϕ ( x ) ϵ F q [ x ] of degree less than q ; thus, the group Ω naturally determines a relation ∼ on F q m × m as follows: if A , B ϵ F q m × m then A ∼ B if ϕ ( A ) = B for some ϕ ϵ Ω . 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 F q m × 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.