This is a doubt in the section 3.3, Induced representations of F&H.
Let $G$ be a group and $H$ is a subgroup and $V$ is a representation of $G$ and $W \subset V$ is a subspace of $V$ which is $H$ invariant. Define $g\cdot W = \{ g\cdot w : w \in W\}$ for all $g \in G.$ Fulton and Harris argue that $g \cdot W$ depend only on the left coset $gH$ of $g$ modulo $H,$ since $$gh \cdot W = g\cdot(h \cdot W) = g \cdot W.$$
My doubt is that $W$ is an $H$ invariant, so $h \cdot W \subseteq W$ but equality is not necessary. Thus, shouldn't it be $$gh \cdot W \subseteq g\cdot W $$ rather than being equal?.
By definition of a group representation, the action of $h$ on $V$ is invertible, so $\dim (h\cdot W) = \dim W$.