Find all normal subgroups of \(A_4\).
Solution
The conjugacy classes of \(A_4\) have sizes \[ 1,\qquad 3,\qquad 4,\qquad 4. \] The class of size \(3\) is \[ \{(12)(34),(13)(24),(14)(23)\}. \] A normal subgroup is a union of conjugacy classes containing the identity, and its order must divide \(12\).
The only possible class-unions with order dividing \(12\) are therefore \[ \{e\},\qquad V_4=\{e,(12)(34),(13)(24),(14)(23)\},\qquad A_4. \] The set \(V_4\) is a subgroup (indeed the Klein four-group), and it is normal because it is a union of conjugacy classes. Hence these are exactly the normal subgroups of \(A_4\).