Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 19I / b

The real projective plane has fundamental group $C_2=⟨a\mid a^2=1⟩$, while the circle has fundamental group $\mathbb{Z}=⟨b⟩$. Apply the Seifert-van Kampen theorem to open neighborhoods of the two wedge summands whose overlap is contractible. This givesChanging the basepoint within this path-connected space gives an isomorphic group.

Define $\boxed{\varphi (a)=(12),\varphi (b)=(123)}$. The relation $a^2=1$ is respected, so the group presentation gives a homomorphism to $S_3$. These permutations generate $S_3$: conjugating $(12)$ by powers of $(123)$ produces the other transpositions. Hence $\varphi$ is surjective.