cohomological dimension of groups vs cohomological dimension of subgroups

Question

Let $\Gamma$ be a group and $\Gamma^\prime$ a subgroup. Then, $\text{cd }\Gamma^\prime \leq \text{cd } \Gamma$ because a projective resolution of $\mathbb{Z}$ over $\mathbb{Z}\Gamma$ can also be regarded as a projective resolution of $\mathbb{Z}$ over $\mathbb{Z}\Gamma^\prime$.

Now, 'Cohomology of Groups' by Brown says that if $\text{cd }\Gamma < \infty$ and $|\Gamma : \Gamma^\prime |< \infty$ (where that denotes the index), then the equality holds.

The proof goes as follows:

It can be shown that if $\text{cd} \Gamma= n$, then there is a free $\mathbb{Z}\Gamma$ module $F$ with $H^n(\Gamma,F)\neq 0$. Let $F^\prime$ be a free $\mathbb{Z}\Gamma^\prime$-module of the same rank. Suppose that $F\cong \bigoplus_{I} \mathbb{Z}\Gamma$ and $F^\prime\cong \bigoplus_{I} \mathbb{Z}\Gamma^\prime$.

Then, $\text{Ind}_{\Gamma^\prime}^\Gamma F^\prime= \mathbb{Z}\Gamma \otimes_{\mathbb{Z}\Gamma^\prime} F^\prime \cong \bigoplus_{I} \mathbb{Z}\Gamma \otimes_{\mathbb{Z}\Gamma^\prime} \mathbb{Z}\Gamma^\prime \cong F$, so by Shapiro's Lemma, $$H^n(\Gamma^\prime, F^\prime)\cong H^n(\Gamma, F)\neq 0.$$ Thus, $\text{cd}\Gamma^\prime \geq n$.

The problem is that I do not see where are we using the hypothesis that $|\Gamma : \Gamma^\prime|<\infty$. Can someone help me, please?

Answer 1

See here for Shapiro's lemma : it holds for induction for homology.

For cohomology, it only holds with coinduction; but if $[\Gamma :\Gamma'] < \infty$ then induction and coinduction coincide, so Shapiro's lemma holds in cohomology with the induced module.