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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:

  1. if $a$ generates $G$ and $b$ generates $H$ then $|a| = |b|$ if G and H are isomorphic.
  2. 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?

Shaun
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Kenneth Winters
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  • Possibly helpful, possible duplicate: https://math.stackexchange.com/questions/2039702/what-is-an-homomorphism-isomorphism-saying/2039715#2039715 – Ethan Bolker Oct 06 '22 at 00:23
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    Part 2 is wrong. A cyclic group $\langle x \rangle $ of order $6$ is generated by the two elements $a=x^2$ and $b=x^3$, but also by the single element $c=x$. – Derek Holt Oct 06 '22 at 07:48

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