Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 7D / ii / Solution

The Cauchy theorem for groups says that if a prime $p$ divides $|G|$, then $G$ contains an element of order $p$.

ConsiderThe first $p-1$ entries determine the last, so $|X|=|G{|}^{p-1}$, which is divisible by $p$. Cyclic rotation acts on $X$: if the product is one, thenEvery orbit has size one or $p$. The fixed points are exactly the constant tuples $(x,\dots ,x)$ satisfying $x^p=1$. Since $|X|\equiv 0(\mathrm{mod}p)$, the number of fixed points is divisible by $p$. The identity supplies one, so there is a nonidentity $x$ with $x^p=1$. Its order divides the prime $p$ and is not one, hence is $p$.