Group Action by Inversion I was reading through the chapter on Cohomology in Dummit and Foote, and I constantly came across the term "action of the cyclic group $C_{2}$ on $G$ by inversion". What does this refer to? Is it an action such that:

$g \cdot e= g \hspace{0.2cm} \forall g \in G \quad g \cdot x=g^{-1} \hspace{0.2cm} \forall g \in G$ 

For $x$ being the non-trivial element in $C_{2}$? This is a very simple question, but I'm not so sure on what is the correct terminology here.
In general, an action of $C_n$ on an object $G$ of a category (for example a group) is a homomorphism $C_n \to \mathrm{Aut}(G)$, which, by the universal property of $C_n = \langle x : x^n = 1 \rangle$, amounts to an element $\sigma \in \mathrm{Aut}(G)$ with $\sigma^n=1$. Therefore, one usually only specifies $\sigma$. The action is then $x^k \mapsto \sigma^k$.
In general, an action of $C_n$ on an object $G$ of a category (for example a group) is a homomorphism $C_n \to \mathrm{Aut}(G)$, which, by the universal property of $C_n = \langle x : x^n = 1 \rangle$, amounts to an element $\sigma \in \mathrm{Aut}(G)$ with $\sigma^n=1$. Therefore, one usually only specifies $\sigma$. The action is then $x^k \mapsto \sigma^k$.