= Center of an upper unitriangular group
{title2=$Z(U_n(q))=\{I+tE_{1n}:t\in\mathbb F_q\},\quad n\geq3$}
The <upper unitriangular group> is generated by adjacent <elementary transvection matrices> $I+sE_{i,i+1}$. Commuting a <matrix> $X$ with each such generator forces every entry above the diagonal to vanish except possibly its $(1,n)$ entry: comparing $XE_{i,i+1}$ with $E_{i,i+1}X$ kills the entries above position $(i,i)$ in column $i$ and to the right of position $(i+1,i+1)$ in row $i+1$. The surviving <matrices> $I+tE_{1n}$ do commute with the entire <group>. In particular the <group center> has order $q$; it distinguishes $U_4(2)$ from $U_3(4)$ despite their common order $64$.
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