function from A5 to A6 
Possible Duplicate:
Homomorphism between $A_5$ and $A_6$ 

Why is it true that every element of the image of the function $f: A_5\longrightarrow A_6$ (alternating groups) defined by $f(x)=(123)(456) x (654)(321)$ does not leave any element of $\{1,2,3,4,5,6\}$ fixed (except the identity)?

I think you are asking why the following is true:

Every element of $A_6$ of the form $(123)(456)x(654)(321)$, with $x$ a nontrivial element of $A_5$, leaves no element of $\{1,2,3,4,5,6\}$ fixed.

This is actually false: let $x = (12)(34)$.  Then $f(x)$ fixes either $5$ or $6$, depending on how you define composition of permutations.
