# Question about Abelian group proof

I prove that if $G$ is Abelian group so if $a,b\in G$ has a finite order so $ab$ has a finite order to.. (Maybe later I'll upload here my proof to see of she is correct....)

Now, I have to show that this is false if the group is not Abelian group with those 2 matrices: $\begin{bmatrix} 0 &-1 \\ 1& 1 \end{bmatrix} and \begin{bmatrix} 0 &1 \\ -1&-1 \end{bmatrix}$

This is the problem:

$\begin{bmatrix} 0 &-1 \\ 1& 1 \end{bmatrix}\cdot \begin{bmatrix} 0 &1 \\ -1&-1 \end{bmatrix}=\begin{bmatrix} 1 &1 \\ -1&0 \end{bmatrix}$

$ord\left(\begin{bmatrix} 1 &1 \\ -1&0 \end{bmatrix}\right)=6$

This is not an infinite order element...

An you have any idea?

Thank you!!

• Any idea about what? You are basically given the solution. take these elements, show that they have finite order, take their product, show it does not have finite order. – Najib Idrissi Nov 8 '13 at 23:56
• @nik - How do I show this: take their product, show it does not have finite order? Thank you! – CS1 Nov 9 '13 at 8:13
• But their product have a finite order - 6... – CS1 Nov 9 '13 at 8:27

You probably saw the group $GL(2,\mathbb R)$, perhaps not under this name. It is the group of all invertible $2\times 2$ matrices with real entries. It is a group under the operation of matrix multiplication. So, this question is probably asking you to identify that the two matrices belong to that group. Then you proceed, a la nik's advise, to compute, in $GL(2,\mathbb R)$, the order of each, the product of the two, and the order of the product.
• If I'll multiply them I'll get:$\begin{bmatrix} 1 &1 \\ -1& 0 \end{bmatrix}$ and the order of this matrix is finite - is 6, so I don't understand how can I show that their product a matrix with infinite order... Thank you! – CS1 Nov 9 '13 at 8:25