Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 4 / c

The trivial action givesbecause $D_n$ is finite. By part b(ii),In the abelianization the relation $aba=b^{-1}$ becomes $b=b^{-1}$. HenceEach finite cyclic group is naturally isomorphic to its character group in $\mathbb{Q}/\mathbb{Z}$, so

Write every element as $a^k{b}^l$ with $k\in \{0,1\}$. The action of this element on $M=\mathbb{Z}$ is multiplication by $(-1{)}^k$. A direct check of the four possibilities for the two exponents of $a$ showsso $\varphi$ is a crossed homomorphism and hence a one-cocycle.

It cannot be principal: if $\varphi (g)=gm-m$ for some $m\in \mathbb{Z}$, then at $g=a$ one would havewhich is impossible in the integers. Thus $[\varphi ]\ne 0$ and