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 [x_{i},x_{j}] = 1, expresses the fact that your group is abelian, the relation (x_{i+1})^{p} = x_{i} expresses the fact that every element of <x_{j}> is a p^{j}-th root of unity.
it should be clear that we have a tower of inclusions:
<x_{1}> ⊂ <x_{2}> ⊂ <x_{3}> ⊂ ......
where <x_{j}> ≅ Z/p^{j} under addition modulo p^{j}.