disproving an isomorphism

**Johngalt13** · April 12th 2012, 05:29 PM

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

Thanks

**Deveno** · April 13th 2012, 03:53 PM

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.

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