What is the total number of all abelian groups of order $675$?

Question

Recently I came across the following question:

What is the total number of all abelian groups of order $675$?

I know how to find out number of abelian groups up to isomorphism. But I have no idea how to approach this question.

If anybody has any idea then please let me know.

Thanks

They had to mean up to isomorphism. Otherwise the question is meaningless, as you can relabel the elements to whatever you want, to whatever cardinality of infinity you want. — Alan, Aug 14 '21 at 09:09
i was thinking the same that the question does not make sense. I can make any number of groups just by relabeling the elements. I wanted to make sure that i am not missing something. Thanks for the reply. — user684646, Aug 14 '21 at 09:13
Can you use the Fundamental Theorem of Finitely Generated Abelian Groups? — Shaun, Aug 14 '21 at 09:22
@shaun That is what we use to find number of abelian groups upto 'isomorphism'. — user684646, Aug 14 '21 at 09:24
And that is the only thing that has sense as stated by @Alan — Augusto Matteini, Aug 14 '21 at 09:26
Yes, @user684646; that's one of its uses and it's why I mentioned it. As others have said, the question has "up to isomorphism" implicitly. — Shaun, Aug 14 '21 at 09:26

Shaun · Accepted Answer · 2021-08-14T09:42:56.260

1

Hint: By the Fundamental Theorem of Finitely Generated Abelian Groups, the number of abelian groups of order $n=p_1^{n_1}\dots p_k^{n_k}$ up to isomorphism is the product of the number of partitions of each of the $n_i$.

edited Aug 14 '21 at 09:42

answered Aug 14 '21 at 09:30

Shaun

47,747

What is the total number of all abelian groups of order $675$?

1 Answers1