I need to prove that no two of any 3 specific groups are isomorphic to each other. I believe I can show this using the fact that generators of a group are preserved under isomorphism but I want to make sure I am understanding what it means for generators to be preserved. Here is my understanding:
- if $a$ generates $G$ and $b$ generates $H$ then $|a| = |b|$ if G and H are isomorphic.
- if $G$ is generated by two elements $a,b$ and $H$ is generated by one element $c$, then $G$ cannot be isomorphic to $H$
Is this a correct understanding of what it means for generators to be preserved under isomorphism?