Cycles of linear permutations over a finite field
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 \(L(x)=\sum^{n-1}_{i=0}a_ ix^{q^ i}\), \(a_ i\in {\mathbb{F}}_ q\).
scientific article; zbMATH DE number 4146071
scientific article