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 $\operatorname{Ind}_H^G(V)$ , is the $\mathbb{Z}[G]$ -module whose underlying vector space is the direct sum $$ \bigoplus_{\sigma \in G/H} \sigma V $$ 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 $$ g(\sigma v) := \tau (h v) $$ 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 $\operatorname{Ind}_H^G(V)$ is independent of the choice of coset representatives $\{g_\sigma\}$ .