Exercise 6, § 1

Let G be a locally compact group having an open and compact subgroup H. Show that for every automorphism $\varphi$ of G, $\mod \varphi \in \mathbf{Q}_+^*$. (Observe that $\varphi(H) \cap H$ has finite index in H and in $\varphi(H)$.) Show that the equality $\Delta_G(s) = 1$ defines an open subgroup of G containing H.