Show that the element $φ(a)\in G'$ has also order d! **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 sorr...--prophetes.ai

Show that the element $φ(a)\in G'$ has also order d! 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 $d$.