Cycles of linear permutations over a finite field
type: Item
instance of: scholarly article
title: Cycles of linear permutations over a finite field
zbMATH Open document ID: 0699.12030
DOI: 10.1016/0024-3795(88)90179-6
author: Gary L. Mullen
author: Theresa P. Vaughan
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: It is well-known that the linear transformations of the finite field \({\mathbb{F}}_{q^ n}\) as a vector space over the subfield \({\mathbb{F}}_ q\) are represented uniquely via substitution by the q-polynomials of the form (1) \(L(x)=\sum^{n-1}_{i=0}a_ ix^{q^ i}\), \(a_ i\in {\mathbb{F}}_{q^ n}\). Moreover, a transformation represented by (1) is nonsingular iff \(\det (a^{q^ j}_{i-j})\neq 0\), where the subscripts are taken modulo n. In this paper the authors study the nonsingular linear transformations on \({\mathbb{F}}_{q^ n}\) which are represented by q-polynomials of the form (1) where the coefficients \(a_ i\) come from the subfield \({\mathbb{F}}_ q\). In particular, they determine the cycle structure of such permutations and they study two notions of equivalence of these linear transformations; namely, a notion \(\sim\) they refer to as ``equivalence'' where \(L_ 1(x)\sim L_ 2(x)\) iff \(L_ 1(x)\) and \(L_ 2(x)\) induce the same cycle structure on \({\mathbb{F}}_{q^ n}\) and a notion \(\approx\) they call ``strong equivalence'' where \(L_ 1(x)\approx L_ 2(x)\) iff \(L_ 1(x)\) and \(L_ 2(x)\) induce the same cycle structure on every T-invariant subspace of \({\mathbb{F}}_{q^ n}\) where T denotes the Frobenius automorphism \(T(\alpha)=\alpha^ q\). They apply their results in the special case of \(q=p^ m\) and \(n=p^ t\) to determine the number of bijective mappings h: \({\mathbb{F}}_{q^ n}\to {\mathbb{F}}_{q^ n}\) commuting with a nonsingular transformation \(L(x)=\sum^{n-1}_{i=0}a_ ix^{q^ i}\), \(a_ i\in {\mathbb{F}}_ q\).
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 4146071
zbMATH Keywords: nonsingular linear transformations
zbMATH Keywords: q-polynomials
zbMATH Keywords: cycle structure
zbMATH Keywords: equivalence
zbMATH Keywords: strong equivalence
zbMATH Keywords: Frobenius automorphism
reviewed by: Joel V. Brawley
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90179-6
OpenAlex ID: W2094792643
cites work: Linear permutation polynomials with coefficients in a subfield
cites work: Q5519913
cites work: Q5690468
cites work: Polynomials Over Finite Fields which Commute with Linear Permutations
cites work: Polynomials and linear transformations over finite fields.
cites work: Linear transformations of a finite field
Wikidata QID: Q127172113
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description: scientific article; zbMATH DE number 4146071