Groups of permutation polynomials over finite fields
Let \(F=\text{GF}(q)\), \(q=p^n >2\) be a finite field and \(a\in F^*\) and \(b\in F\) and \(1\leq k \leq q-2.\) Let \(\tau_{a,b}: x \to ax+b\), and \(\pi_k:x\to x^k\) the linear polynomial and the power map, where \(k\) is prime to \(q-1.\) Define \(G_k=\langle \tau_{a,b}, \pi_k \mid a\in F^*, b\in F \rangle\). Then (i) If \(k=p^i\) and \(d=\text{gcd}(n,i)\), then \(G_k\) is the semidirect product of the affine group \(\text{AGL}(1,F)\) and the subgroup of order \({{n}\over{d}}\) generated by the semilinear map \(\pi_{p^d}.\) (ii) If \(p\) is odd and \(k\) is not a power of \(p,\) then \(G_k\) is the symmetric group, \(\text{Sym}(F).\) (iii) If \(p=2\) and \(k\) is not a power of \(2,\) then \(G_k \supseteq \text{Alt}(F).\) Moreover, \(G_k=\)Sym\((F)\) if and only if \(\pi_k\) is an odd permutation. This is a generalization of a theorem of \textit{L. Carlitz} [Proc. Am. Math. Soc. 4, 538 (1983; Zbl 0052.03704)] on symmetric groups over a finite field.
scientific article; zbMATH DE number 1229769
scientific article