Exercise 10, § 6

Let $G$ and $H$ be two groups, $u:H\to G$ a homomorphism, $M$ a $\mathbf Z^{(G)}$-module. For every $q\geq 0$, one deduces from the homomorphisms $\alpha$ and $\beta$ of Exercise 1 homomorphisms

which are simply denoted by $u_q$ and $u^q$ if there is no ambiguity concerning $M$.

a) For $f\in C^q(G,M)$, denote by $f_u$ the element of $C^q(H,M)$ such that

for $h_1,\ldots,h_q\in H$. Prove that the mapping $f\mapsto f_u$ is a morphism of complexes from $C^q(G,M)$ into $C^q(H,M)$, which induces on homology the homomorphism $\bigoplus_q u^q$. Define analogously a morphism from $C^q(H,M)$ into $C^q(G,M)$ which induces $(\bigoplus u_q)$.

b) Suppose that $H=G$ and take for $u$ an inner automorphism of $G$. Prove that the homomorphisms $u_q$ and $u^q$ are equal to the identity (reason by induction on $q$).