Proposition 2

Let G be a group operating on a set E and, for all $x \in E$, let $S_x$ be the stabilizer of x. For all $\alpha \in G$, $S_{\alpha x} = \alpha S_x \alpha^{-1}$.

If $s \in S_x$, then $\alpha s \alpha^{-1}(\alpha x) = \alpha s x = \alpha x$, whence $\alpha S_x \alpha^{-1} \subset S_{\alpha x}$. As $x = \alpha^{-1}(\alpha x)$, $\alpha^{-1} S_{\alpha x} \alpha \subset S_x$, whence $S_{\alpha x} \subset \alpha S_x \alpha^{-1}$.

It is seen similarly that, if A and B are two subsets of E and T is the transporter (resp. strict transporter) of A to B, then the transporter (resp. strict transporter) of $\alpha A$ to $\alpha B$ is equal to $\alpha T \alpha^{-1}$.