Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 1 / a

The integers are closed under addition, addition is associative, zero is an identity, and $-n$ is the additive inverse of $n$. Thus $(\mathbb{Z},+)$ is an abelian group.

The nonzero integers are closed under multiplication and contain the identity, but most elements have no inverse in the set: for example, the multiplicative inverse of $2$ is $1/2\notin {\mathbb{Z}}^{\ast }$. Hence $({\mathbb{Z}}^{\ast },\times )$ is not a group.

Composition is associative, andagain has nonzero slope. The identity is $(1,0)$ andThus these maps form the real affine group of the line, a nonabelian group.