Show that there does not exist a group homomorphism from G onto G'

Question

Show that there does not exist a group homomorphism from G onto G', where G and G' are groups of order 9 and 6 respectively

Answer 1

Alternatively, any group of even order, like $G'$, has a element of order $2$. Show that this can't be in the image of $G$ for any homomorphism.

If $\phi(g)^2=1_{G'}$ in $G'$, then $1_{G'}=\phi(1_G)=\phi(g^9)=\phi(g)^8\cdot \phi(g)= \phi(g)$.

Answer 2

For any homomorphism between finite groups, the elements that are hit are all hit equally many times. In this case that would mean $1.5$ times, which is absurd.

Answer 3

The order of the image of a homomorphism between finite groups must divide the order of the domain but $6$ does not divide $9$.