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Math Help - How many Abelian groups are there

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    How many Abelian groups are there

    Let p be a prime number. How many Abelian groups (up to isomorphism) are there of order p^100.

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    Quote Originally Posted by mandy123 View Post
    Let p be a prime number. How many Abelian groups (up to isomorphism) are there of order p^100.

    I am stuck, Please Help
    If the problem was about p^3 then the Fundamental Theorem for Abelain groups would say it is one of the following:

    \mathbb{Z}_{p^3}
    \mathbb{Z}_{p^2}\times \mathbb{Z}_p
    \mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_p

    What about your more general problem?
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    I don't have a more general problem, this is all I am given. Is that bad?

    So if I follow what you did, would there be 100 Abelian groups?
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    Quote Originally Posted by mandy123 View Post
    I don't have a more general problem, this is all I am given. Is that bad?

    So if I follow what you did, would there be 100 Abelian groups?
    note that 100 = 5^2 \cdot 2^2

    so that if we let p = 5 and q = 2, then in the most basic cases you have

    \mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2}

    now you want all possible combinations of those two in direct products. the number of combinations you come up with is the number of Abelian groups
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    Quote Originally Posted by Jhevon View Post
    note that 100 = 5^2 \cdot 2^2

    so that if we let p = 5 and q = 2, then in the most basic cases you have

    \mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2}

    now you want all possible combinations of those two in direct products. the number of combinations you come up with is the number of Abelian groups
    I think mandy is talking about p^{100}.
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    Quote Originally Posted by mandy123 View Post

    Let p be a prime number. How many Abelian groups (up to isomorphism) are there of order p^100.

    I am stuck, Please Help
    by the fundamental theorem of finite abelian groups, the answer is the number of partitions of 100, which according to this website is equal to 190569292.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    I think mandy is talking about p^{100}.
    oh! oh yes, i see. i thought it was groups of order 100 as opposed to p^{100}
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    by the fundamental theorem of finite abelian groups, the answer is the number of partitions of 100, which according to this website is equal to 190569292.
    there should be some kind of combination formula for this right? similar to the one used in the multinomial theorem
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    Quote Originally Posted by Jhevon View Post
    there should be some kind of combination formula for this right? similar to the one used in the multinomial theorem
    I do not think there is a partitions formula.
    The # of partitions is a very complicated combinatorics problem.

    There are ways to get them using recurrence relations and all that stuff. But as far as a formula that give you an answer it is does not exist. At least I never seen one. . I am scared!
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