= Finite groups act freely on suitable closed orientable surfaces
{title2=$g=1+|G|(h-1)$}
If a finite <group> $G$ is generated by $h$ elements, send the $a$-generators of the genus-$h$ <surface group> to those generators and its $b$-generators to the identity. The surface relation is satisfied, so this is a surjective <group homomorphism>. Its <kernel> defines a connected regular <covering space>, on which the <deck transformation group> $G$ acts freely. The lifted <orientation> and multiplicativity of the <Euler characteristic> give genus $g=1+|G|(h-1)$.
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