Question

find all normal subgroup of A4​

Answer 1

Answer:

Up to changing name to the symbols, the fact that (234)−1(123)(234)=(134)(234)−1(123)(234)=(134)is enough, since it proves that any element of order 33 is conjugated to another element which is not one of its powers.

Here is a non-elementary (but more "adaptable") proof: since the Sylow 33-subgroups are pairwise conjugated, if there was a normal subgroup of order 33 it would be the unique Sylow 33-subgroup, i.e. the unique subgroup of order 33, and this is false because A4A4 has more than one subgroup of order 33.