Abstract
In this paper, we investigate the sign of permutations induced by the Anick automorphism and the Nagata-Anick automorphism over finite fields. We shall prove that if the Anick automorphism and the Nagata-Anick automorphism are defined over a prime field of characteristic two, they induce odd permutations, and otherwise, they induce even permutations.
Highlights
IntroductionAffn(K), EAn(K)) the set of polynomial automorphisms
We investigate the sign of permutations induced by the Anick automorphism and the Nagata-Anick automorphism over finite fields
We shall prove that if the Anick automorphism and the Nagata-Anick automorphism are defined over a prime field of characteristic two, they induce odd permutations, and otherwise, they induce even permutations
Summary
Affn(K), EAn(K)) the set of polynomial automorphisms Let us denote by TAn(K) the subgroup of GAn(K) generated by two subgroups Affn(K) and EAn(K). When we restrict the map πq to GAn(Fq), πq (G) is a subgroup of Sym(Fnq) for any subgroup G ⊆ GAn(Fq). If there exists F ∈ GAn(F2m ) such that sgn (π2m (F)) = −1, we must have F ∈ GAn(F2m ) \ TAn(F2m ). This indicates that the polynomial automorphism F is wild. For q = 2m and m ≥ 2, do there exist polynomial automorphisms such that the permutations induced by the polynomial automorphisms belong to Sym(Fnq) \ Alt(Fnq)?. It is natural to consider the sign of the famous polynomial automorphisms such as the Nagata automorphism (Nagata, 1972), the Anick automorphism (Cohn, 2006), and the Nagata-Anick automorphism (Cohn, 2006)
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