Prove that $A_4$ has no normal subgroup of order $3.$
This is how I started: Assume that $A_4$ has a normal subgroup of order $3$, for example $K$. I take the Quotient Group $A_4/K$ with $4$ distinct cosets, each of order $3$. But I want to prove that these distinct cosets will not contain $(12)(34),(13)(24)$ and $(14)(23)$> Therefore a contradiction. Please help, I'm really stuck!!