Unitriangular group
= Unitriangular group
{title2=$\operatorname{UT}_n(F)$}
The <unitriangular group> $\operatorname{UT}_n(F)$ 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 inverse
$$
(I_n+N)^{-1}=I_n-N+N^2-\cdots+(-1)^{n-1}N^{n-1}.
$$
Thus these <matrices> form a <subgroup> of the <general linear group>. The case $\operatorname{UT}_3(\mathbb F_3)$ 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> $\mathbb F_3$.