I have been studying the induced representation of a finite group, in particular the form given by the following definition:
Given an $H$ representation, $(\mu, W)$,
$Ind_H^GW$ $= \{f:G \rightarrow W | f(gh) = \mu(h^{-1})f(g)$ $ \forall g \in G, h \in H \}$ .
I am told that it follows from this that every element of the induced representation is determined by the values it takes on the cosets of H in G, so that the induced representation has dimension $[G:H] dim(W)$. I am struggling to see how these functions are determined by their values on the cosets and would appreciate some help with this.
Well if $g\in g_iH$ then there is some $h$ such that $g= g_ih$ and so $f(g) = f(g_ih) = \mu(h^{-1})f(g_i)$.
This means that if $\displaystyle\bigsqcup_{i\in I} g_iH$ is a coset decomposition of $G$, and if $f$ and $f'$ coincide on the $g_i$'s then they coincide on all of $G$.
Using the same notations as above, this gives a linear isomorphism $$f\mapsto (f(g_i))_{i\in I}$$ which yields the desired dimension