How do you show that the only nontrivial normal subgroup of $A_4$, which is also not the whole group is the Klein 4 group, denoted by $V$ (or isomorphic to the Klein 4 group)?
I've shown before that $V$ is a normal subgroup of $S_4$, and that $V \subset A_4$. Is there a way to use those facts?
Best Answer
The following may help you:
1) A subgroup of a group is normal in it iff it is a union of conjugacy classes
2) Two permutations in $\,S_n\,$ are conjugate iff they have the same cycle decomposition (i.e., the same lengths of cycles and the same ammount of cycles of each length)
3) A conjugacy class of an even permutation in $\,S_n\,$ remains exactly the same class in $\,A_n\,$ unless all the disjoint cycles in the cyclic decomposition of the permutation are of odd and different lengths, in which case the equivalence class splits in two classes in $\,A_n\,$