= Unitriangular group of degree three over F3
{title2=$UT_3(\mathbb F_3)$}
The <upper unitriangular group> $UT_3(\mathbb F_3)$ consists of $M(a,b,c)=\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\end{pmatrix}$ for $a,b,c\in\mathbb F_3$. Its multiplication is $M(a,b,c)M(a',b',c')=M(a+a',b+b',c+c'+ab')$. The elements $M(1,0,0)$ and $M(0,1,0)$ do not commute. Yet every element is $I+N$ with $N^3=0$, so characteristic three gives $(I+N)^3=I$. It is a nonabelian <finite group> of order $27$ whose nonidentity elements all have order three.
Back to article page
