Let $\displaystyle f : G -> G'$ be a homomorphism. Let $\displaystyle a$ be an element of $\displaystyle G$ of period 10. Prove that $\displaystyle f(a)$ must have period 1 or 2 or 5 or 10.

I can prove that $\displaystyle f(a)$ may have period 10.

$\displaystyle a^{10} = e$ because $\displaystyle a$ have period 10

$\displaystyle f(a^{10}) = [f(a)]^{10} = e'$ by property of homomorphism

Therefore $\displaystyle f(a)$ may have period 10.