# disproving an isomorphism

• Apr 12th 2012, 05:29 PM
Johngalt13
disproving an isomorphism
Attachment 23585

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

Thanks
• Apr 13th 2012, 03:53 PM
Deveno
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.