There are two elements $x, y$ in a group $(G, *)$ such that every element in the group can be written as a product of some number of x's and y's in some order. It is known that $$x * x = y * y = x * y * x * y = y * x * y * x = e$$ where $e$ is the identity element. The maximum number of elements in such a group is _________.

We are given that $G = \langle x, y \rangle$ with relations:

From $(xy)^2 = e$:

Multiplying both sides by $y$ on the right (and using $y^2 = e$):

Multiplying both sides by $x$ on the left (and using $x^2 = e$):

Since $x$ and $y$ commute, any word generated by $x$ and $y$ can be simplified to the form $x^a y^b$.

Since $x^2 = e$ and $y^2 = e$, the exponents $a$ and $b$ can only take values in $\{0, 1\}$.

Thus, the distinct elements of $G$ can only be:

This gives a maximum of 4 elements (isomorphic to the Klein four-group $V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$).