# Math Help - disproving an isomorphism

1. ## disproving an isomorphism

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

Thanks

2. ## 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.