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In homework of the book 'Introduction to algebra and finite fields', I have a question about finite group.

There is the question:

Suppose $G$ is a finite Abelian group. For any $a,b\in G$, there exists $c\in G$ such that $o(c)=[o(a),o(b)]$. o(c) is the order of element c, and [o(a), o(b)] is the lcm symbol.

How to prove that?

Thanks for you help!

Greg Martin
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zkp_learner
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  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Jun 05 '22 at 06:19
  • This isn't true as stated: consider if $a=b^{-1}$ for example. – Greg Martin Jun 05 '22 at 06:39
  • @GregMartin What isn't true as stated? Note that we aren't claiming $c=ab$. – Arthur Jun 05 '22 at 07:37
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    You posted this earlier, https://math.stackexchange.com/questions/4465439/finite-abelian-group-question and instead of trying to improve it, you just posted it, unchanged, again. That's an abuse of this website. Please don't do that. – Gerry Myerson Jun 05 '22 at 07:40
  • @GerryMyerson Sorry, that question had been closed, I don't know how to make it open. – zkp_learner Jun 05 '22 at 07:55
  • @gerryMyerson I have changed my question statement. – zkp_learner Jun 05 '22 at 07:58
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    The way to (try to) reopen it is to read what you were told: "Closed. This question does not meet Mathematics Stack Exchange guidelines. It is not currently accepting answers.... Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." and to follow the links that accompanied that message. – Gerry Myerson Jun 05 '22 at 09:01
  • @Arthur Well if $c$ is unrelated to $a$ and $b$ then the statement definitely isn't true. – Greg Martin Jun 05 '22 at 18:49
  • @GregMartin That's not what the problem statement says either. Modulo a little imperfect English (which I think we ought to forgive in any international community) it says $\forall a \forall b\exists c\big(o(c) = [o(a), o(b)]\big)$. That's not $c$ unrelated to $a, b$, but it isn't a fixed relationship either. – Arthur Jun 05 '22 at 18:58
  • I truly did not understand that that was the question being asked. @Arthur, next time you could help out by editing the post itself. – Greg Martin Jun 05 '22 at 20:12
  • @GregMartin Well, that's the problem, isn't it? I didn't see the need. It's difficult to judge from a comment like your original complaint whether more fault for this miscommunication should be placed with you for reading a bit too fast, or with the author for writing unintelligibly. I understood it just fine, so I just assumed the former (which I do myself all the time, so it's easy for me to assume of others). – Arthur Jun 07 '22 at 12:52

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