Results 1 to 2 of 2

Thread: Torsion

  1. #1
    Junior Member
    Joined
    Jun 2011
    Posts
    34

    Torsion

    [IMG]file:///C:/Users/JC/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
    If
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22

    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 $\displaystyle \displaystyle \text{Tor}(G)= \bigoplus_{p}\mathbb{Z}_p$--this is easy. To prove ii) prove that for every $\displaystyle \displaystyle (x_p)\in G$ and $\displaystyle n=p_1^{\alpha_1}\cdots p_n^{\alpha_n}$ you can find $\displaystyle \displaystyle (y_p)$ such that $\displaystyle (x_p)-n(y_p)\in\text{Tor}(G)$. To do this factorize $\displaystyle n=p_1^{\alpha_1}\cdots p_m^{\alpha_m}$. Then, note that for $\displaystyle p\ne p_1,\cdpts,p_m^{\alpha}_m$ you have that $\displaystyle n\in\mathbb{Z}_p^\times$. Define then $\displaystyle (y_p)$ to be such that $\displaystyle y_p=0$ if $\displaystyle p\in\{p_1,\cdots,p_m\}$ and $\displaystyle [n]_p^{-1}x_p$ otherwise. To prove three note that if $\displaystyle G/\text{Tor}(G)$ were a direct summand of $\displaystyle G$ then there would be an exact sequence $\displaystyle 0\to G/\text{Tor}(G)\to G$, and what's the problem with that (hint: $\displaystyle G$ is HIGHLY non-divisible).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Torsion 2
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: Dec 6th 2011, 03:07 PM
  2. Torsion
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Dec 11th 2010, 02:17 PM
  3. Torsion of the curve
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Oct 26th 2009, 04:09 AM
  4. Torsion
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Oct 25th 2009, 08:43 AM
  5. Non-torsion elements
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Jun 8th 2009, 11:44 PM

Search Tags


/mathhelpforum @mathhelpforum