4.3 Three actions of a group on its elements

It is mentioned here.

Definition

Let $g\in G$. Define 3 permutations in $\mathrm{Sym}(G)$ by

$$\begin{array}{rl}\check{g}:x& \mapsto xg\\ \check{g}:x& \mapsto gx\\ \tilde{g}:x& \mapsto g^{-1}xg\end{array}$$

with the corresponding groups $\check{G},\check{G}$ and $\tilde{G}$.

Lemma

For any group $G$, the following equations hold:

• $N_{\mathrm{Sym}(G)}(\check{G})=\check{G}:\mathrm{Aut}(G)$
• $\check{G}{C}_{\mathrm{Sym}(G)}(\check{G})=\check{G}\circ \check{G}=\check{G}:\mathrm{Inn}(G)$