[Math] Show two groups are isomorphic

Question

I need to show two groups are isomorphic. Since I know they are of the same order, would finding an element that generates the other elements, in both groups, suffice to show that they are isomorphic?

Answer 1

Yes. If you can find an element $x$ which generates the finite group $G$ then it is cyclic. If $G=\langle x\rangle$ and $G'=\langle x'\rangle$ are both cyclic, and $|G|=|G'|$ then $G$ is isomorphic to $G'$ by an isomorphism $\phi\colon G\to G'$ which is defined by $\phi(x^k)=x'^k$. Show that this map is an isomorphism.