Show that $H$ $=$ {$(id), (12)(34), (13)(24), (14)(23)$} is the only non-trivial normal subgroup for $A_4$.
Looking at this question, what I started with was:
By Lagrange's theorem, there are only subgroups of order: 1, 2, 4, 6 and 12. Clearly the trivial normal subgroups are $A_4$ and ${(id)}$ thus we can eliminate the subgroups of order 1 and 12.
From what I proved from an assignment a month ago, there doesn't exist a subgroup of order 6 for $A_4$ so we can eliminate that as well.
Now we have to deal with the subgroups of order 2 and order 4. I'm having a block here now. I know the condition of a normal subgroup, but I can't really think of a way to show that H is the only non-trivial normal subgroup.
Any hints? :O