For a group represented as $\langle x,y\mid x^4,y^5,xyx^{-1}y\rangle$, how to determine its precise order? I guess I may need to use the universal property, but how to construct functions to determine size?
The question does have some steps or hints like the show it has at most 20 elements first, then construct some subjective to get its minimal order, etc.
I wonder what is the standard procedure to solve such questions
For instance, in this particular case, first the third "anticommutativity relation" allows you to write every element in the form $a^ib^j$. That together with the first two relations tells us that the order is at most $20$.
Next note that the elements of a certain semi-direct product $\Bbb Z_5\rtimes_\varphi\Bbb Z_4$ satisfy the given relations. Namely the one with $\varphi:\Bbb Z_4\to\rm{Aut}\Bbb Z_5\cong\Bbb Z_4$ given by $\varphi(1)=2$.
This implies that there is a surjective homomorphism from $G$ to this group. Thus we have a lower bound of $20$ for the order as well.