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?
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.