TITLE: Transitivity of representation induction
QUESTION [5 upvotes]: Let $K\subset H\subset G$ be some triple of finite groups and $T: K\longrightarrow \mathrm{GL}(V)$ - some representation f $K$. We are to prove the transitivity of induction: $Ind_K^G(V)\simeq Ind_H^G(Ind_K^H(V))$. I would like to do it somehow like that:
First, we identify $Ind_K^G(V)$ with the tensor product $\mathbb{C}[G]\otimes_{\mathbb{C}[K]}V$ (it is $\mathbb{C}[G]$-module: $g:g'\otimes v\mapsto gg'\otimes v$). Thus, I need to show that $\mathbb{C}[G]\otimes_{\mathbb{C}[K]}V$ is isomorphic (as $\mathbb{C}[G]$-module) to $\mathbb{C}[G]\otimes_{\mathbb{C}[H]}(V\otimes_{\mathbb{C}[K]} \mathbb{C}[H])$. Is it possible to do it compactly, could you help?

REPLY [3 votes]: Induction is left adjoint to restriction. The restriction $G \rightarrow K$ is the composition $(H \rightarrow K) \circ (G \rightarrow H)$, so its left adjoint is the composition of the left adjoints.