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Math Help - presentetion of Prüfer group

  1. #1
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    Red face presentetion of Prüfer group

    How would you prove that [itex] < x_1,x_2 ... | [ x_i , x_j ] =1, i,j \in N , x_1 ^ p = 1, x_{i+1} ^p = x_i , i \in N > [/itex] is presentation of [itex] Z_{p^ \infty} [/itex]
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  2. #2
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    Re: presentetion of Prüfer group

    How would you prove that  < x_1,x_2 ... | [ x_i , x_j ] =1, i,j \in N , x_1 ^ p = 1, x_{i+1} ^p = x_i , i \in N > is presentation of   Z_{p^ \infty}
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  3. #3
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    Re: presentetion of Prüfer group

    what is you DEFINITION of the Prüfer group?

    the commutator relation [xi,xj] = 1, expresses the fact that your group is abelian, the relation (xi+1)p = xi expresses the fact that every element of <xj> is a pj-th root of unity.

    it should be clear that we have a tower of inclusions:

    <x1> ⊂ <x2> ⊂ <x3> ⊂ ......

    where <xj> ≅ Z/pj under addition modulo pj.
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