uniqueness of induced representation I am studying the book "Representation Theory" by Fulton and Harris. And I just can not understand the part where they prove the uniqueness of induced representation. If someone could explain it I'd greatly appreciate it! It's on page 33 and it goes:
Choose a representative $g_{\sigma} \in G$ for each coset $\sigma \in G/H,$ with $e$ representing the trivial coset $H$. To see the uniqueness, note that each element of $V$
has a unique expression $v = \sum g_{\sigma} w_{\sigma}$ for elements $w_{\sigma}$ in $W$. 
Given $g \in G$ write $g \centerdot g_{\sigma} =  g_{\tau} \centerdot h$ for some $\tau \in G/H$ and $h \in H.$ Then we must have 
$$g \centerdot (g_{\sigma} w_{\sigma})= g_{\tau}( h w_{\sigma})\;.$$
This proves The uniqueness ...
I understand everything until the end, but I just don't understand how this proves the uniqueness... If someone could give me a little more explanation, I would appreciate it!
Thanks!
 A: The question of uniqueness is whether we have any freedom in defining the action of an arbitrary element $g$ on $W$. The equation shows that we don't: The action of $g$ on each summand of $W$, and thus on $W$, is entirely determined by the action of $H$ on $V$. Writing out the action for a linear combination and denoting $g_\tau$ by $g_{\sigma g}$ and $h$ by $h_g$ to mark the dependencies, we have
$$
g\sum g_\sigma w_\sigma=\sum g(g_\sigma w_\sigma)=\sum g_{\sigma g}(h_gw_\sigma)\;,
$$
and this fully determines the action of $g$ on any element of $V$.
A: Given a group action $H \mapsto \mathrm{GL}(W)$, they want to extend to a group action $G \mapsto \mathrm{GL}\left( \sum_{\sigma \in G/H} \sigma W \right)$ acting on the direct sum over coset spaces.  
Is the action of $g \in G$ well-defined on any element, $v = \sum g_{\sigma} w_{\sigma}$ ? Fulton's identity says we can always find a coset representative $g \centerdot g_{\sigma} = g_{\tau} h \in g_\tau H$.  The coset is unique but the coset representative is only unique up to conjugacy by elements of $H$.
The action of $g$ on $g_\sigma w_\sigma$ decomposes the action of $h: W \mapsto W$ and the action of $g_\tau$ permuting the various coset spaces $W_\sigma$.
