I am reading Humprheys' Algebraic Groups, stuck at an apparently simple point
In section 8.2 Actions of Algebraic groups(line 6, paragraph 1), The transporter is defined: Let $Y ,Z$ be subsets of $X$ (An affine variety of on which $G$ is acting regularly)
Define: $\text{Tran}_{G}(Y, Z) = \{ x \in G \mid x.Y \subset Z \}$.
The claim: $\text{Tran}_{G}(Y, Z)$ is a submonoid! I don't see why this is true, if $a, b \in \text{Tran}_{G}(Y, Z)$ then $a.Y \subset Z$ and $b.Y \subset Z$ then why should $ab.Y = a.(b.Y) \subset Z$?
$b.Y$ need not be in Y!
Please give some hint.