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Math Help - disproving an isomorphism

  1. #1
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    disproving an isomorphism

    disproving an isomorphism-abalg.png

    Why aren't these two groups isomorphic? They both have the same number of elements with the same orders

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  2. #2
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    Re: disproving an isomorphism

    the group \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3 is abelian, but in \mathrm{SL}_3(\mathbb{Z}_3), if:

    A = \begin{bmatrix}1&1&0\\0&1&2\\0&0&1 \end{bmatrix}, B = \begin{bmatrix}1&1&0\\0&1&1\\0&0&1 \end{bmatrix}, then

    AB = \begin{bmatrix}1&2&1\\0&1&0\\0&0&1 \end{bmatrix}, BA = \begin{bmatrix}1&2&2\\0&1&0\\0&0&1 \end{bmatrix},

    so this group is non-abelian.
    Last edited by Deveno; April 13th 2012 at 04:12 PM.
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