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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]
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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