Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 5 / a (Matrices over the field with three elements)

In the finite field ${\mathbb{Z}}_3=\{0,1,2\}$, perform ordinary addition and multiplication and then take the remainder modulo $3$. The tables areThe nonzero elements have multiplicative inverses $1^{-1}=1$ and $2^{-1}=2$.

Parametrize the matrices byDirect matrix multiplication givesThis proves closure. The identity matrix is $h(0,0,0)$, and the formulagives both a left and a right inverse, as substitution in the product formula verifies. Thus each element is an invertible matrix, and the subgroup test shows that $H$ is a subgroup of ${\mathrm{GL}}_3({\mathbb{Z}}_3)$. It is the unitriangular group ${\mathrm{UT}}_3({\mathbb{F}}_3)$ and has $3^3=27$ elements.

The identity is the identity matrixLeft or right matrix multiplication by it preserves every entry, as is also immediate from the product formula above.

Take $x=h(1,0,0)=I_3+E_{12}$ and $y=h(0,0,1)=I_3+E_{23}$. Their matrix products areThey differ in the $(1,3)$ entry because $E_{12}{E}_{23}=E_{13}$ whereas $E_{23}{E}_{12}=0$. Therefore $xy\ne yx$, and $\boxed{H\text{ is a non-abelian group}}$.