= Sharply three-transitive on a projective line
{title2=$PGL_2(F)\text{ acts regularly on ordered distinct triples of }\mathbb P^1(F)$}
For any field $F$, a <projective linear group> element is uniquely determined by its images of three distinct points of the <projective line>, and any ordered distinct target triple is attainable. Given representatives $v_z,v_x$ forming a <basis> and $v_y=A v_z+B v_x$, the <matrix> with columns $A v_z,B v_x$ sends $(0,1,\infty)$ to $(x,y,z)$. Distinctness forces $A,B\ne0$. The <matrices> realizing one projective map differ by a nonzero scalar; over $\mathbb F_p$ there are $p-1$ such <matrices>.
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