Unitriangular group (${\mathrm{UT}}_n(F)$)

The unitriangular group ${\mathrm{UT}}_n(F)$ consists of upper triangular matrices over a field $F$ with every diagonal entry equal to one. Every such matrix is $I_n+N$ for a strictly upper triangular matrix $N$. Products have the same form, and $N^n=0$ gives the inverseThus these matrices form a subgroup of the general linear group. The case ${\mathrm{UT}}_3({\mathbb{F}}_3)$ is noncommutative but every element has cube equal to the identity matrix: $(I_3+N{)}^3=I_3$ because $N^3=0$ and $3=0$ in the finite field ${\mathbb{F}}_3$.