Linear fractional group as Galois group

Abstract

We compute all signatures of $PS{L}_2({\mathbb{F}}_7)$, and $PS{L}_2({\mathbb{F}}_{11})$ which classify all orientation preserving actions of the groups $PS{L}_2({\mathbb{F}}_7)$, and $PS{L}_2({\mathbb{F}}_{11})$ on compact, connected, orientable surfaces with orbifold genus $\ge 0$. This classification is well-grounded in the other branches of Mathematics like topology, smooth, and conformal geometry, algebraic categories, and it is also directly related to the inverse Galois problem.