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Math Help - Elementary abelian group

  1. #1
    Senior Member abhishekkgp's Avatar
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    Elementary abelian group

    Find the number of subgroups of index p of the elementary abelian group E_{p^n}.
    The answer should be \frac{p^n-1}{p-1} but i don't see how to approach this problem.
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    MHF Contributor Drexel28's Avatar
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    Re: Elementary abelian group

    Quote Originally Posted by abhishekkgp View Post
    Find the number of subgroups of index p of the elementary abelian group E_{p^n}.
    The answer should be \frac{p^n-1}{p-1} but i don't see how to approach this problem.
    Note first that E_{p^n} is a vector space over \mathbb{F}_p. Moreover, if H\leqslant E_{p^n} then it's also trivially a subspace of E_{p^n} since it's closed under addition. Thus, the subgroups of index p, which are precisely the subgroups of order p^{n-1}, which are precisely the subspaces of dimension n-1. Now, using the common formula that the number of subspaces of an n-dimensional \mathbb{F}_q-space is \displaystyle \frac{(q^n-1)(q^n-q)\cdots(q^n-q^{k-1})}{(q^k-1)(q^{k}-q)\cdots(q^k-q^{k-1})} (which is just a simple counting argument about choosing linearly independent subsets of size k--it's on page 412 of Dummit and Foote for example) and pluggin in k=n-1 gives the desired result.
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    Senior Member abhishekkgp's Avatar
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    Re: Elementary abelian group

    Quote Originally Posted by Drexel28 View Post
    Note first that E_{p^n} is a vector space over \mathbb{F}_p. Moreover, if H\leqslant E_{p^n} then it's also trivially a subspace of E_{p^n} since it's closed under addition. Thus, the subgroups of index p, which are precisely the subgroups of order p^{n-1}, which are precisely the subspaces of dimension n-1. Now, using the common formula that the number of subspaces of an n-dimensional \mathbb{F}_q-space is \displaystyle \frac{(q^n-1)(q^n-q)\cdots(q^n-q^{k-1})}{(q^k-1)(q^{k}-q)\cdots(q^k-q^{k-1})} (which is just a simple counting argument about choosing linearly independent subsets of size k--it's on page 412 of Dummit and Foote for example) and pluggin in k=n-1 gives the desired result.
    thank you drexel but your solution is too advanced for me... actually i wanted to solve the question given on pg 168(question 8(b)) of dummit and foote. i had posted the question on this thread http://www.mathhelpforum.com/math-he...reply&t=186533 . then i took a special case of this problem which is "prove that number of subgroups of order p in E_{p^n} is equal to the number of subgroups of index p in E_{p^n}." Now i was able to prove that E_{p^n} has \frac{p^n-1}{p-1} subgroups of order p.

    I want to solve the question using only the material covered till pg. 168. Your solution uses Vector spaces which is given on pg. 388 and i have not yet read that..
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    MHF Contributor Drexel28's Avatar
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    Re: Elementary abelian group

    Quote Originally Posted by abhishekkgp View Post
    thank you drexel but your solution is too advanced for me... actually i wanted to solve the question given on pg 168(question 8(b)) of dummit and foote. i had posted the question on this thread http://www.mathhelpforum.com/math-he...reply&t=186533 . then i took a special case of this problem which is "prove that number of subgroups of order p in E_{p^n} is equal to the number of subgroups of index p in E_{p^n}." Now i was able to prove that E_{p^n} has \frac{p^n-1}{p-1} subgroups of order p.

    I want to solve the question using only the material covered till pg. 168. Your solution uses Vector spaces which is given on pg. 388 and i have not yet read that..
    Use the fourth isomorphism theorem to reduce the problem to the case of elementary abelian groups and then take a look at my solution. I mean, not to be pushy, but the only alternative I can think of is a nasty argument using the Orbit Stabilizer theorem. It's easier to just use what I said, and realize that math isn't an insular subject--subjects are supposed to be intermingled.
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    MHF Contributor Drexel28's Avatar
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    Re: Elementary abelian group

    If you're unwaivering in your desire to do it 'by the book' a LONG solution can be found here.
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    Senior Member abhishekkgp's Avatar
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    Re: Elementary abelian group

    Quote Originally Posted by Drexel28 View Post
    Use the fourth isomorphism theorem to reduce the problem to the case of elementary abelian groups.
    I am not sure how to do that. Can you drop a hint?
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    MHF Contributor Drexel28's Avatar
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    Re: Elementary abelian group

    Quote Originally Posted by abhishekkgp View Post
    I am not sure how to do that. Can you drop a hint?
    In the link I provided the guy does the same thing I would do. Try reading it.
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