= Involution orbits in an affine group of odd prime degree
{title2=$x\mapsto-x+b:\quad1\text{ fixed point},\ (\ell-1)/2\text{ pairs}$}
A nonidentity <involution> in the <affine group over a finite field> $\mathbb F_\ell$, for odd prime $\ell$, has linear coefficient $-1$: a translation has order $\ell$, while $u^2=1$ forces $u=\pm1$. Its unique fixed point is $b/2$, and its remaining orbits have size two. This orbit count computes <primes in an intermediate field as double cosets> for an order-two <decomposition group> acting on the radical roots.
// Target: number-theory.bigb
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