# Symmetric genus of a finite group

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The symmetric genus of a finite group $G$, denoted $\sigma(G)$, is defined in the following equivalent ways:
1. It is the smallest genus $\sigma$ of a compact connected oriented surface on which $G$ acts faithfully via diffeomorphisms, which may be orientation-preserving or orientation-reserving.
2. It is the smallest genus $\sigma$ of a compact connected Riemann surface on which $G$ acts faithfully via Riemann surface isomorphisms or anti-isomorphisms, i.e., by mappings that are either conformal or anti-conformal (i.e., they reverse the roles of $i,-i$).
3. it is the smallest genus $\sigma$ of a compact connected two-dimensional Riemannian manifold on which $G$ acts faithfully via isometries of the Riemannian metric.