Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 7 / 2 / i / Solution

Identify $G$ with a subgroup of $\mathrm{Sym}(\Omega )$ and adjoin one new point $\infty$, fixed by every element of $G$. A one-point extension of a permutation group is a transitive subgroup $G^{+}\le \mathrm{Sym}(\Omega \cup \{\infty \})$ whose point stabilizer at $\infty$ is exactly this embedded copy of $G$. Equivalently, one may use an isomorphic permutation representation with a distinguished point whose stabilizer acts on the remaining points as the original group.

Since $G$ is transitive on $\Omega$, the extension is doubly transitive group action. The orbit-stabilizer theorem gives the useful necessary order condition