Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 2 / 11E / c / Solution

For the general linear group over ${\mathbb{F}}_{31}$, choose a nonzero first column in $31^2-1$ ways and a second column outside its span in $31^2-31$ ways. ThusSince $5^3=125\equiv 1(\mathrm{mod}31)$ and $5\ne 1$, the residue $5$ has order three. The nine diagonal matrices $\mathrm{diag}(5^i,5^j)$, with $0\le i,j<3$, form a subgroup isomorphic to $C_3\times C_3$. It is an abelian group and a Sylow subgroup for $p=3$. Every other Sylow subgroup for $p=3$ is conjugate to it by the Sylow theorems, hence is also Abelian.