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Math Help - Cyclic Subgroup

  1. #1
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    Cyclic Subgroup

    I need to prove that H={(1 n 0 1) | n is an element of the integers} is a cyclic subgroup of GL2(reals). H is a matrix with the first row 1 and n and the second row 0 and 1. I appreciate the help!
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  2. #2
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    Re: Cyclic Subgroup

    i suppose you mean:

    H = \left\{\begin{bmatrix}1&n\\0&1 \end{bmatrix} | n \in \mathbb{Z} \right\}

    the way to do this is show that:

    \phi:H \to \mathbb{Z} defined by:

    \phi\left(\begin{bmatrix}1&n\\0&1 \end{bmatrix}\right) = n

    is an isomorphism of groups. this means that if:

    A = \begin{bmatrix}1&k\\0&1 \end{bmatrix};\ B = \begin{bmatrix}1&m\\0&1 \end{bmatrix}

    you need to show that:

    \phi(AB) = \phi(A) + \phi(B)

    and that \phi is bijective.
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