TITLE: Does a homomorphic image of even permutations consist of even permutations?
QUESTION [6 upvotes]: If $f:S_n \to S_n$ is a homomorphism, prove $f(A_n) \subseteq A_n$.

If every image of a transposition is even, then there is nothing to prove, but it is not sure..
How can I prove the problem?

REPLY [5 votes]: $A_n$ is the set of products of an even number of transpositions  

this is one common definition of $A_n$

$A_n$ is the set of products of products of two transpositions  

just group pairs of transpositions

$A_n$ is the set of products of 3-cycles  

$(12)(23)=(123)$, $(12)(34)=(123)(234)$

$A_n$ is the set of products of squares  

$(123)=(132)^2$ and the square of every permutation is even

$f$ maps products of squares to products of squares  

$f$ is a homomorphism

$f$ maps $A_n$ into $A_n$.