Bordered and unbordered Klein surfaces with maximal symmetry (Q1077548)

An \(M^*\)-group G is a group of automorphisms of a compact Klein surface [see \textit{N. L. Alling} and \textit{N. Greenleaf}, Foundations of the theory of Klein surfaces (1971; Zbl 0225.30001)] of algebraic genus \(g\geq 2\) such that \(\| G\| =12(g-1)\), i.e. is maximal [see \textit{C. L. May}, Pac. J. Math. 59, 199-210 (1975; Zbl 0422.30037)]. In the present work the authors exhibit four new infinite families of simple \(M^*\)-groups, and determine with the aid of a computer the groups PSL(n,q) of order less than 50.000 that are \(M^*\)-groups. Using these results, the authors prove the existence of seven topologically different Klein surfaces S of algebraic genus \(g=1013\), all of them having maximal symmetry, i.e. \(\| Aut S\| =12(g-1)\).