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Math Help - Prove that the following is a subgroup

  1. #1
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    Prove that the following is a subgroup

    Let p be a prime. Let G be a finite abelian group such that xp=e for all X in G.

    1) Let k be a positive integer, and let a1,...,ak be elements of G such that the products a1i1 ... akik (the 1 and k on the i's should be subscripts) are distinct for all choices of integers i1 ... ik from 0 through p-1. Prove that the set of elements (a1...ak) is a subgroup of G.

    2) Prove that

    (0,0,0), (1,1,0), (1,0,1), (0,1,1)

    is a subgroup of Z2 X Z2 X Z2 isomorphic to Z2 X Z2.

    3) Let G be the direct product of Zp X ... X Zp of m copies of Zp for a positive integer m. Let H be a subgroup of G that has more than one element. Prove that H is isomorphic to the direct product Zp X ... X Zp for an integer k <= (less than or equal to) m. Use Exercise 1.
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  2. #2
    MHF Contributor
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    Re: Prove that the following is a subgroup

    Hey TimsBobby2.

    It's been a little while since I did this stuff, but can you show us what you've tried, thought about, and any attempts (both partial and full)?
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