Let \(q\) be a prime power, \(n\geq 3\) be an odd integer, and \(m=(q^n+1)/(q+1)\). Let \(K=\mathbb F_{q^{2n}}\). The generalized GK function field \(\mathcal C_n=K(x,y,z)\) is defined by the equations \[ x^q+x=y^{q+1},~ y^{q^2}-y=z^m. \] The aim of this paper is to determine the automorphism group \(G\) of this function field.
Let \(\mathcal H_n=K(x,y)\) and \(\mathcal X_n=K(y,z)\), which are described by the first and second of the above equations respectively. Then \(\mathcal C_n\) is the compositum of these two fields. Denote by \(P_{\infty}\) the common pole of \(x,y\) and \(z\) in \(\mathcal C_n\), and let \(G(P_{\infty})\) be the subgroup of \(G\) consisting of all elements that fix \(P_{\infty}\). The authors first determine this subgroup \(G(P_{\infty})\), and then show that \(P_{\infty}\) is fixed by every element of \(G\). Thus \(G=G(P_{\infty})\). The authors also obtain the order of \(G\), which is \(q^3(q-1)(q^n+1)\).
The arguments involved in this paper is fairly explicit and elementary. It is worth-noting that R. Guralnick et al. [J. Algebra 361, 92–106 (2012; Zbl 1285.11094)] also determined the automorphism group of \(\mathcal C_n\) independently, using a very different method based on some group-theoretical arguments.