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Math Help - Torsion

  1. #1
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    Torsion

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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Torsion

    Quote Originally Posted by jcir2826 View Post
    [IMG]file:///C:/Users/JC/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
    If
    I don't know what i) says, presumably that \displaystyle \text{Tor}(G)= \bigoplus_{p}\mathbb{Z}_p--this is easy. To prove ii) prove that for every \displaystyle (x_p)\in G and n=p_1^{\alpha_1}\cdots p_n^{\alpha_n} you can find \displaystyle (y_p) such that (x_p)-n(y_p)\in\text{Tor}(G). To do this factorize n=p_1^{\alpha_1}\cdots p_m^{\alpha_m}. Then, note that for p\ne p_1,\cdpts,p_m^{\alpha}_m you have that n\in\mathbb{Z}_p^\times. Define then (y_p) to be such that y_p=0 if p\in\{p_1,\cdots,p_m\} and [n]_p^{-1}x_p otherwise. To prove three note that if G/\text{Tor}(G) were a direct summand of G then there would be an exact sequence 0\to G/\text{Tor}(G)\to G, and what's the problem with that (hint: G is HIGHLY non-divisible).
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