The order of the Quaternion Group

I'm trying to find an easy way to prove the order of the Quaternion Group is a non-abelian group of order 8 without many computations, anyone has an idea?

Note we can define the Quaternion Group as the group generated by the following matrices $$A= \left( \begin{matrix} 0 & 1 \\ -1 & 0\\ \end{matrix} \right)$$ $$B= \left( \begin{matrix} 0 & i \\ i & 0\\ \end{matrix} \right)$$ Thanks

Solution 1:

Or maybe you can find the following an interesting point, a very similar to Ittay's simple way: $$Q_8=\langle a,b\mid a^4=1, a^2=b^2, ba=a^{-1}b\rangle=\{1,a,b,a^2,ab,a^3,a^2b,a^3b\}$$ Note that $ba=a^{-1}b$ shows it is non abelian.

Solution 2:

Depending on the definition you have in mind, the following might work for you: $ij=k$ while $ji=-k$ shows that it is not abelian. Counting the number of elements in $\{\pm 1, \pm i, \pm j, \pm k\}$ shows that it is a group of order $8$.