Originally Posted by

**jackie** Thank you so much for your help, Taluivren.

I think I'm still stuck on the second part of 1). I tried to show $\displaystyle n$ divides $\displaystyle m$ where $\displaystyle n=ord(g)$ and $\displaystyle m=ord(\phi(g))$, but I was unable to show this. I also tried to write n=mk since m divides n by previous result, but could not get anywhere.

But I think I got the second part of 2). I proved that $\displaystyle {\phi}^{-1}(x)$ generates G, where x is a generator of $\displaystyle \phi(G)$.

Now I'm stuck on another homomorphism property. Show that if H is a subgroup of G with $\displaystyle ord(H)=n$, then $\displaystyle ord(\phi(H)$ divides n. Any help is very much appreciated.