The author proves that there are two isomorphism classes of groups of order 21: the class of the cyclic group $C_{21}$, and the class of a group $G$ generated by two elements $x$ and $y$ that satisfy the relations $x^7 = 1, y^3 = 1, yx=x^2y.$
Why is this product map surjective?
Because it is an injective map from a set with $21$ elements into another set with $21$ elements.