Show Composition Map $S_3 \to S_4/V$ is an isomorphism.

Question

Consider the subset $V = \{1,(12)(34),(13)(24),(14)(23)\}$ of $S_4$. Let $\pi : S_4 \to S_4/V$ be the quotient map. There is a natural group homomorphism $\rho : S_3 \to S_4$, which sends a 2-cycle $(i j)$ to $(i j)$, and a 3-cycle $(i j k)$ to $(i j k)$. Show that the composition map $\pi \rho : S_3 \to S_4/V$ is an isomorphism.

Answer 1

Obviously $\pi \rho$ is a group homomorphism as a composition of group homomorphism. Now prove that this map is surjective, which shouldn't be difficult to conclude just by looking at the cosets of $V$ in $S_4$. Now to prove that the map is injective just prove that $\ker \pi \rho$ is $\{e\}$, which should be easy by looking at which elements are in $\ker \pi$ and which of these elements are in the image of $\rho$.