Challenge 25: Finite Abelian Groups

  • #1
18,077
7,497

Main Question or Discussion Point

What is the smallest positive integer n such that there are exactly 3 nonisomorphic Abelian group of order n
 

Answers and Replies

  • #2
2,788
586
Its 8!
 
  • #3
18,077
7,497
Last edited:
  • #4
2,788
586
Show you're work! :)
I thought its legitimate to use our searching skills!:D
 
  • #5
34,039
9,882
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.
 
  • #6
94
10
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:
 
  • #7
473
13
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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