induced representation

Let $G$ be a group, $H\subset G$ a subgroup, and $V$ a representation of $H$, considered as a $\mathbb{Z}[H]$–module. The induced representation of $\rho$ on $G$, denoted ${\mathrm{Ind}}_H^G(V)$, is the $\mathbb{Z}[G]$–module whose underlying vector space is the direct sum

$$\underset{\sigma \in G/H}{\oplus }\sigma V$$

of formal translates of $V$ by left cosets $\sigma$ in $G/H$, and whose multiplication operation is defined by choosing a set ${\{g_{\sigma }\}}_{\sigma \in G/H}$ of coset representatives and setting

$$g(\sigma v):=\tau (hv)$$

where $\tau$ is the unique left coset of $G/H$ containing $g\cdot g_{\sigma }$ (i.e., such that $g\cdot g_{\sigma }=g_{\tau }\cdot h$ for some $h\in H$).

One easily verifies that the representation ${\mathrm{Ind}}_H^G(V)$ is independent of the choice of coset representatives $\{g_{\sigma }\}$.