Super G-spaces

We review the basic theory of super $G$-spaces. We prove a theorem relating the action of a super Harish-Chandra pair $(G_0,\mathfrak{g})$ on a supermanifold to the action of the corresponding super Lie group $G$. The theorem was stated in [DM99] without proof. The proof given here does not use Frobenius theorem but relies on Koszul realization of the structure sheaf of a super Lie group (see [Kosz83]). We prove the representability of the stability subgroup functor.