Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 19I / i / Solution

Let $|G|=p^2$. By irreducible character degree divides the group order, each irreducible degree divides $p^2$. A degree is at most $\sqrt{|G|}=p$ by the sum of squares of irreducible degrees. If a degree $p$ occurred, its square would already consume the entire sum $p^2$, leaving no room for the trivial character. Hence every irreducible degree is one. A finite group has only linear irreducible characters exactly when it is abelian, so every group of order $p^2$ is abelian.