Let $G$ be an affine algebraic Group, $H \subseteq G$ a closed subgroup. The Normalizer of $H$ is usually defined as $\text{N}_G(H) := \{ g \in G | gHg^{-1} = H\}$.
However, some authors define it as $\text{Tran}_G(H,H) = \{g \in G | gH \subseteq H \}$
(for example Def. 4.14 in "Actions and Invariants of Algebraic Groups" by Santos, Rittatore).
How are these definitions equivalent?
Edit: The definition of Transporters involves a variety $X$ on which $G$ acts morphically. For subsets $Y, Y \subseteq X$ the transporter is then given by $\text{Tran}_G(Y,Z) = \{g \in G | gY \subseteq Z \}$.
I suspected that G acts on itself via plain group multiplication so the definition makes sense for a subgroup $H \subset G$.
The definition of transporter you quote is
In this definition, $g \cdot Y$ refers to the action of $G$ on $X$ (and its subsets) which is implicit in the statement that $X$ is a $G$-variety, the action beging a map \begin{align*} &G \times X \to X\\ &(g, x) \mapsto g \cdot x \end{align*} I am using the notation $g \cdot x$ for the action of $g \in G$ on $x \in X$, to differentiate this from the product within $G$, which I denote by juxtaposition.
In this case, regard $G$ as a $G$-variety, the action being given by $$ g \cdot x = g x g^{-1} $$ for $g, x \in G$. (On the left side we have the action, on the right side a group product.) With this action, the transporter is nothing else but the normaliser.