Unique involution in SL2 over an odd field
= Unique involution in SL2 over an odd field
{title2=$A^2=I,\ \det A=1\Longrightarrow A=\pm I$}
Over a field of odd characteristic, an involutory <matrix> has <minimal polynomial> dividing $(X-1)(X+1)$, so it is diagonalizable with <eigenvalues> in $\{1,-1\}$. In dimension two, <determinant> one requires either both <eigenvalues> one or both minus one. Hence the only element of order two in $SL_2$ is $-I$; the identity has order one.