A Note of Equivalence Classes of Matrices over a Finite Field

J. V. Brawley, Gary L. Mullen

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1 Scopus citations

Abstract

Let [formula omitted] denote the algebra of mxm 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 [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.

Original languageEnglish (US)
Pages (from-to)279-287
Number of pages9
JournalInternational Journal of Mathematics and Mathematical Sciences
Volume4
Issue number2
DOIs
StatePublished - 1981

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)

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