Let be a finite abelian group and let be a prime. Prove that the number of subgroups of of order equals the number of subgroups of of index .

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- Aug 22nd 2011, 05:42 AMabhishekkgpFinite abelian group
Let be a finite abelian group and let be a prime. Prove that the number of subgroups of of order equals the number of subgroups of of index .

- Aug 22nd 2011, 02:10 PMDevenoRe: Finite abelian group
although this is not "quite" the same problem, i believe it is similar enough to help you: finite abelian group

- Aug 22nd 2011, 08:57 PMModusPonensRe: Finite abelian group
If you have Hungerford's Algebra, look at the theorem II.2.6.(iii). Since A (or G in the book's notation) is finite, the F goes away. Now count the subgroups of index p and the subgroups of order p.

- Aug 22nd 2011, 11:42 PMabhishekkgpRe: Finite abelian group
- Aug 24th 2011, 08:29 AMabhishekkgpRe: Finite abelian group
I can prove that has a total of distinct subgroups of order .

To solve the original question(post #1) by the approach you have suggested i need to find the number of subgroups( of course distinct) of order of abelian groups like etc. What i have found out (which i am not 100% sure is correct) that number of subgroups of order of is same as the number of subgroups of order of etc. is this correct??

Now i couldn't figure out how to find the number of subgroups having*index*. Can you please help on this one?? - Aug 24th 2011, 06:49 PMModusPonensRe: Finite abelian group
I have to say that I thought I knew the answer to this problem when I posted the hint. But I realised later that I didn't. Anyway, I can say that has more subgroups of order p than because the latter has only , and , while the first has at least and

If I would have to resolve this problem, I would try to find, with specific examples, such as , the group of index 5 for each group of order 5 and thus understand the one to one correspondence. - Aug 24th 2011, 08:12 PMabhishekkgpRe: Finite abelian group
- Aug 25th 2011, 06:29 PMModusPonensRe: Finite abelian group
You're right. Sorry.

I don't have the time necessary to devote myself to the problem, so I will leave it as it is. - Aug 25th 2011, 06:48 PMDrexel28Re: Finite abelian group
If interested, I provided the necessary tools to solve this in this thread.