= Unipotent conjugacy in SL2 over a finite field
{title2=$U_a\sim U_b\ \Longleftrightarrow\ b/a\text{ is a nonzero square}$}
For a <finite field> of odd characteristic and nonzero $a,b$, let $U_a=\begin{pmatrix}1&a\\0&1\end{pmatrix}$. An arbitrary determinant-one matrix $P=\begin{pmatrix}r&s\\t&u\end{pmatrix}$ satisfies $PU_a=U_bP$ only if $t=0$ and $ar=bu$. The determinant condition $ru=1$ then gives $b/a=r^2$. Conversely $\operatorname{diag}(r,r^{-1})U_a\operatorname{diag}(r,r^{-1})^{-1}=U_{ar^2}$. Thus nonzero upper <unipotent matrices> fall into two <conjugacy classes> under this square-class criterion. For example, $U_1$ and $U_3$ are conjugate over $\mathbb F_{11}$ because $5^2=3$, but not over $\mathbb F_5$. The criterion compares these particular upper unipotent representatives; it does not classify all determinant-one matrices.
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