Challenge 25: Finite Abelian Groups

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Discussion Overview

The discussion revolves around identifying the smallest positive integer n such that there are exactly 3 nonisomorphic Abelian groups of order n. The scope includes theoretical exploration and mathematical reasoning related to group theory.

Discussion Character

  • Exploratory, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant claims that the answer is 8, suggesting that n cannot be square-free and needs factors that are multiples of each other to yield multiple non-isomorphic groups.
  • Another participant agrees with the answer of 8 but requests to see the work behind the conclusion.
  • Some participants express a playful tone regarding the use of searching skills to arrive at answers.
  • A later reply indicates a misunderstanding of the challenge, with a participant initially thinking it asked for 3 non-Abelian groups, leading to a different answer.

Areas of Agreement / Disagreement

There is some agreement that 8 is the smallest n, but the discussion includes playful challenges and a misunderstanding that introduces uncertainty about the correct interpretation of the problem.

Contextual Notes

Participants discuss the properties of n, particularly its need to not be square-free, but the mathematical steps and definitions remain unresolved.

Greg Bernhardt
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What is the smallest positive integer n such that there are exactly 3 nonisomorphic Abelian group of order n
 
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Shyan said:
Its 8!
Show your work! :)
 
Last edited:
Greg Bernhardt said:
Show you're work! :)
I thought its legitimate to use our searching skills!:D
 
n cannot be square-free (it needs factors that are multiples of each other), otherwise you don't get multiple non-isomorphic groups. The first two numbers with that property are 4 (leading to 2 different groups, corresponding to "4" and "2x2") and 8 ("8", "4x2", "2x2x2"). Therefore, 8 is the smallest n.
 
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Shyan said:
Its 8!

I thought its legitimate to use our searching skills
Oh come on! :H:nb):oldeek: If that's not a big spoiler, I don't know what is.:oldeyes:
 
Haha, I misread the challenge as asking for 3 non-Abelian groups, so - also using searching skills - I came to a different answer.
 

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