Math Help - presentetion of Prüfer group

1. 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 $Z_{p^ \infty}$

2. 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 $Z_{p^ \infty}$

3. 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.