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Math Help - Non-torsion elements

  1. #1
    MHF Contributor Swlabr's Avatar
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    Non-torsion elements

    Do these form a subgroup? That is to say, as the inverse of a non-torsion element (an element of non-finite order) is non-torsion, basically the question is: If we multiply two non-torsion elements together do we get another non-torsion element?
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  2. #2
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    Is the identity a nontorsion element?
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  3. #3
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    siclar has already answered the question. a less trivial question is to find examples of two torsion elements a,b of a group such that ab becomes non-torsion.
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    Quote Originally Posted by NonCommAlg View Post
    siclar has already answered the question. a less trivial question is to find examples of two torsion elements a,b of a group such that ab becomes non-torsion.
    A= \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) and B= \left( \begin{array}{cc} 0 & 1 \\ -1 & -1 \end{array} \right) then A^4 = I = B ^3 but AB has infinite order.
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by siclar View Post
    Is the identity a nontorsion element?

    Sorry - do the non-torsion elements along with the identity form a subgroup?
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  6. #6
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    Quote Originally Posted by Swlabr View Post
    Sorry - do the non-torsion elements along with the identity form a subgroup?
    still no! for example in the group of 2 \times 2 invertible matrices over \mathbb{Q} let a=\begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix} and b=\begin{pmatrix}1 & 0 \\ -2 & 1 \end{pmatrix}. then both a,b are non-torsion but ab=\begin{pmatrix}-1 & 1 \\ -2 & 1 \end{pmatrix} is torsion because (ab)^4=I.
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