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

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.