I'm reading out of "Differential Forms in Algebraic Topology" by Bott and Tu. In the introductory chapter they define the fundamental group. I understood their definition until they claim the following:
$\pi_1\left(X,p\right)$ is generated by six elements $\left\{x_1,x_2,x_3,y_1,y_2,y_3\right\}$ subject to the single relation $$\prod_{i=1}^3 \left[x_i,y_i\right] =1 $$ Where $\left[x_i,y_i\right]$ is the commutator $x_iy_ix_i^{-1}y_i^{-1}$ and 1 is the identity.
I struggle to see how this is true. Is there a proof of this somewhere that I can read or is there a simple fact that I'm overlooking?