Harvey-Wiman hypermaps (Q1924357)

The author shows the following: Let \((\alpha ,\sigma)\) be a hypermap of genus \(g\geq 2\), \(p\) be the smallest divisor of \(\text{ Aut}(\alpha ,\sigma)\) and let \(\psi\) be an automorphism of \((\alpha ,\sigma)\). Then (i) either \(o(\psi )=p(1+2g/(p-1))\), where \(p\) and \(1+2g/(p-1)\) are coprime and the quotient hypermap with respect to the subgroup of order \(p\) is planar, (ii) or \(o(\psi )\leq 2pg/(p-1)\). This is an improvement of a corresponding Theorem of \textit{W. J. Harvey} [Q. J. Math., Oxford II. Ser. 17, 86-97 (1966; Zbl 0156.08901)]\ concerning an automorphism of a compact Riemann surface. He also characterizes hypermaps admitting an automorphism of order \(p(1+2g/(p-1))\) and \(2pg/(p-1)\), respectively.