One-point extension of a permutation group
= One-point extension of a permutation group
Adjoin a new point $\omega$ fixed by a transitive group $G$. A one-point extension is a transitive group on the enlarged set whose stabilizer of $\omega$ is exactly $G$, acting on the remaining points in the prescribed way. Its order is $(n+1)|G|$ in the finite degree-$n$ case. The <double-coset criterion for a one-point extension> gives a concrete generator test.