Originally Posted by

**jackie** How can I show that there is no homomorphism from $\displaystyle S_3$ to $\displaystyle Z_3$?

Suppose there is one homomorphism $\displaystyle f$. Let $\displaystyle g \in S_3$, then $\displaystyle f(g)=0$ or $\displaystyle f(g)=1$ or $\displaystyle f(g)=2$. I know $\displaystyle ord(f(g))$ must divide $\displaystyle ord(g)$. I have $\displaystyle ord(0)=\infty$, $\displaystyle ord(1)=3$, and $\displaystyle ord(2)=2$. I want to show that there is an element in $\displaystyle S_3$ such that $\displaystyle ord(f(g))$ does not divide $\displaystyle ord(g)$. I'm stuck on this because based on my approach, $\displaystyle ord(f(g))$ can be 1, and 1 divides anything.

Can anyone give me help here?