0
$\begingroup$
  1. 1
  2. 3
  3. 7
  4. 11

I did not understand the meaning of question specifically the statement "up to isomorphic" means? Also is their any set formula to find number of abelian groups?

2 Answers 2

1

$1001=7\times 11 \times 13$, since this number is square free the only abelian group of order $1001$ is cyclic.

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    Does the statement 'up to isomorphic' mean That group of order 1001 becomes isomorphic to some abelian group? – 2017-01-20
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    it means that if you have two isomorphic groups you don't count them twice – 2017-01-20
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    What if the number would be 1000? – 2017-01-20
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    then it is harder, since $1000=2^3 \times 5^3$ you want $P(3)\times P(3)=3^2=9$, where $P$ is the partition function. You may want to look up the fundamental theorem of finite abelian groups. – 2017-01-20
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    P(3)=3 because 3 can be expressed in 3 ways, {3,2+1,1+1+1} ryt?..and so P(4)=5 – 2017-01-20
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    yes exactly.${}{}{}$ – 2017-01-20
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    Okay, thank u so much – 2017-01-20
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    happy to help. ${}{}{}{}$ – 2017-01-20
2

Use the prime factorisation of 1001.