Is there an epimorphism from $A_9$ to $\mathbb{Z}_{20}$

Question

Is there an epimorphism from $A_9$ to $\mathbb{Z}_{20}$

I've tried with elements with different orders and trying to show that there is no element in $A_9$ such that $o(f(x))|o(x)$ ($f$ is epimorphism), but it did not work.

How to do this task?

Answer 1

There's no element of order 20 in $A_9$. There are such elements in $S_9$, but you should be able to see that they all have the same cycle structure as $(12345)(6789)$, which is an odd permutation.

Answer 2

Suppose $A_9\to\mathbb{Z}_{20}$ is an epimorphism with kernel $K$, then $\mathbb{Z}_{20}\cong A_9/K$.

$A_n$ is simple for $n\ge 5$ so $K$ is trivial or $A_9$. Both cases are clearly impossible, so no such epimorphism exists.