EXERCISE:
Consider that $Φ:G\rightarrow G'$ homomorphism and $a\in G$ which has order d.
Show that the element $φ(a)\in G'$ has also order d.
I am sorry i could't appose any attempt here but i don't even know how to start.I have just started to examine homomorphism and i have to clear my mind on them!
It would be very helpful if someone can give me some hints or a thourough solution to my problem.
I want to learn how i can work with these problems as i don't have many experience with these type of problems!
Thanks in advance!
To expound on user8734617's comment:
We have that $a^d =e$ (where $e$ is the identity). Therefore $\phi(a)^d = \phi(a^d) = \phi(e) =e$.
However, all this tells us is that the order of $\phi(a)$ divides $d$.