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Math Help - Proper subgroups of fraction group

  1. #1
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    Proper subgroups of fraction group

    Let p be a prime and let  \mathbb {Z} (p^ \infty ) = \{ \hat { \frac {a}{b} } \in \frac { \mathbb {Q} }{ \mathbb {Z} } : a,b \in \mathbb {Z} \ , \ b=p^i \ , \ i \geq 0 \} .
    Prove that the only proper subgroups of  \mathbb {Z} (p^ \infty ) are the finite cylic groups J_k = < \hat { \frac {1}{p^k} } >

    Proof so far.

    Suppose to the contrary that H is a subgroup of  \mathbb {Z} (p^ \infty ) such that  H \neq \mathbb {Z} (p^ \infty ) and  H \neq  < \hat { \frac {1}{p^k} } > \ \ \ \forall k . Now, suppose that  H = < \frac {a}{b}> , then \frac {a}{b} \neq \frac {1}{p^k} . But how would I get a contradiction? Thanks.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let p be a prime and let  \mathbb {Z} (p^ \infty ) = \{ \hat { \frac {a}{b} } \in \frac { \mathbb {Q} }{ \mathbb {Z} } : a,b \in \mathbb {Z} \ , \ b=p^i \ , \ i \geq 0 \} .
    Prove that the only proper subgroups of  \mathbb {Z} (p^ \infty ) are the finite cylic groups J_k = < \hat { \frac {1}{p^k} } >
    let J be a proper subgroup of \mathbb{Z}_{p^{\infty}}. then there exists an integer n \geq 1 such that \frac{1}{p^n} + \mathbb{Z} \notin J. let m=\min \{n \geq 1: \ \frac{1}{p^n} + \mathbb{Z} \}. then \frac{a}{p^n} + \mathbb{Z} \in J, for all a \in \mathbb{Z} and 0 \leq n \leq m-1.

    in here i showed that \frac{a}{p^n} + \mathbb{Z} \notin J, for all integers a coprime with p and for all n \geq m. therefore: J=\{\frac{a}{p^n}+\mathbb{Z}: \ \ a \in \mathbb{Z}, \ 0 \leq n \leq m-1 \} = <\frac{1}{p^{m-1}} + \mathbb{Z} >. \ \Box
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