TITLE: The alternating group is a normal subgroup of the symmetric group
QUESTION [5 upvotes]: For an exercise, I need to prove that the alternating group $A_n$ is a normal subgroup of the symmetric group $S_n$.
As clue they say we can use a group homomorphism $\operatorname{sgn} : S_n \to \{-1,1\}$. I really don't see how i can use this.... can somebody help?

REPLY [2 votes]: Add another method to prove in addition to Arpit's answer: conjugation preserves cycle type; so $s a s^{-1} \in A_n$