I Wording of a group action

1. Mar 30, 2017

Mr Davis 97

My textbook says the following: "Let $G$ be a group and $G$ act on itself by left conjugation, so each $g \in G$ maps $G$ to $G$ by $x \mapsto gxg^{-1}$". I am confused by the wording of this. $g$ itself is not a function, so how does it map anything at all? I am assuming this is supposed to mean that $g \cdot x = gxg^{-1}$ is the definition of the action, but why does it use the map notation as if $g$ is a function, when $g$ is really an element?

2. Mar 30, 2017

Staff: Mentor

It's $\varphi : G \longrightarrow Inn(G)$ with $g \longmapsto \iota_g$ where $\iota_g : x \longmapsto gxg^{-1}$ is the inner automorphism "conjugation by $g$". $\varphi$ is a group homomorphism from $G$ to the group of inner automorphisms which is a subgroup of the automophism group $Aut(G)$ of $G$. Whether you say "$G$ acts on $X$" or $G \longmapsto Aut(X)$ is a group homomorphism is the same thing. The notation "acts / operates on" with a dot is only a bit shorter than to introduce this homomorphism $\varphi$ and replace the dot by $\varphi$.