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$\begingroup$

a,b are elements of the group G

I have no idea how to even start - I was thinking of defining a,b as two square matrices and using the non-commutative property of matrix multiplication but I'm not sure if that's the way to go...

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    Matrix multiplication just happened to be the first thing I thought of but I don't know if that's actually how I'm supposed to approach it. And thanks for clarifying, I thought that might be the reason.2012-11-10

2 Answers 2

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Since you want this to be in a group, take the group of permutations of $\{1,2,3\}$ and take the counterexample $a$ and $b$ to be two distinct transpositions, say $(1,2)$ and $(2,3)$.

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    @wj32 No, it hasn't been covered yet but I'll have a look at your link.2012-11-10
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Consider the multiplicative subgroup of the quaternions consisting of $\pm 1$, $\pm i$, $\pm j$, and $\pm k$. Let $a=i$ and $b=j$. We have $ij^2=-i$ and $j^2i=-i$ but $ij\ne ji$.