Prove that characteristic subgroups are normal. Give an example of a normal subgroup that is not characteristic.
Suppose is characteristic, and let denote conjugation by . We know that is an automorphism of ; since is characteristic, we have for all . Hence is normal.
Now define by and , and extend homomorphically to all of . Note that , so that is surjective; since is finite, is an automorphism.
Recall that every subgroup of is normal. Now ; thus is normal but not characteristic in .