Cycles of linear permutations over a finite field

Gary L. Mullen, Theresa P. Vaughan

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


We study the cycle structure of those permutations of the finite field Fqn of the form L(x) = ∑ i=0 n-1aixqi where each ai ε{lunate} Fq. For such a permutation, the problem of finding its cycle decomposition of Fqn can be reduced to finding its cycle decomposition on certain T-invariant subspaces of Fqn, where T is the operator defined by T : x → xq. If L1(x) and L2(x)M are in the above form, we say that L1(x) and L2(x) are equivalent if L1(x) and L2(x) induce the same cycle decomposition of Fqn, and we say they are strongly equivalent if they induce the same cycle decomposition in every T-invariant subspace of Fqn. We show that these notions are not the same, and we give characterizing theorems for each.

Original languageEnglish (US)
Pages (from-to)63-82
Number of pages20
JournalLinear Algebra and Its Applications
Issue numberC
StatePublished - Sep 1988

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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