Attachment 23585
Why aren't these two groups isomorphic? They both have the same number of elements with the same orders
Thanks
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Attachment 23585
Why aren't these two groups isomorphic? They both have the same number of elements with the same orders
Thanks
the group $\displaystyle \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3$ is abelian, but in $\displaystyle \mathrm{SL}_3(\mathbb{Z}_3)$, if:
$\displaystyle A = \begin{bmatrix}1&1&0\\0&1&2\\0&0&1 \end{bmatrix}, B = \begin{bmatrix}1&1&0\\0&1&1\\0&0&1 \end{bmatrix}$, then
$\displaystyle AB = \begin{bmatrix}1&2&1\\0&1&0\\0&0&1 \end{bmatrix}, BA = \begin{bmatrix}1&2&2\\0&1&0\\0&0&1 \end{bmatrix}$,
so this group is non-abelian.
