Definition of induced representation

[kof9595995]

http://planetmath.org/encyclopedia/InducedRepresentation.html
The thing I don't get is the definition of a group element $g$'s action on a vector $\sigma v$. In the link it defines the action as $g(\sigma v):=\tau (hv)$, where $\tau$ is the unique left coset of G/H containing $gg_{\sigma}$(i.e. such that $gg_{\sigma}=g_{\tau}h$ for some h belonging to H).
The thing that confuses me is the arbitrariness of h, i.e., if $gg_{\sigma}=g_{\tau}h$, we can always have $gg_{\sigma}=g_{\tau}h_1h_1^{-1}h=(g_{\tau}h_1)(h_1^{-1}h)$, where $h_1$ is some arbitrary element of H. Now $g_{\tau}h_1$ still belongs to coset $\tau$, but $h_1^{-1}h$ will be a different element in H, say $h_1^{-1}h=h'$ , then according to the previous definition of group action, we have $g(\sigma v):=\tau (h'v)$. So what shall I make of this arbitrariness?

 

[kof9595995]

Ok I think I figured out my misunderstanding, once a set of representatives of cosets is chosen, you can't change it, so if $g_{\sigma}$ is a representative, then $g_{\sigma}h_1$ can't be on the list of representatives.