Question 60

Given a group $(G, *)$ generated by elements $x, y$ satisfying:

1. Deduce Commutativity:

   • Since $x^2 = e$, $x^{-1} = x$.
   • Since $y^2 = e$, $y^{-1} = y$.
   • From $(xy)^2 = e$:
     $$x * y * x * y = e \implies x * y = (x * y)^{-1} = y^{-1} * x^{-1} = y * x.$$
   • Hence, $x * y = y * x$, which means the generators commute with each other.

2. Determine the Elements of $G$:

   • Since $x$ and $y$ commute, any word formed by combinations of $x$ and $y$ can be simplified to the canonical form $x^a y^b$.
   • Since $x^2 = e$ and $y^2 = e$, the exponents $a$ and $b$ can only take values in $\{0, 1\}$.
   • This yields at most $2 \times 2 = 4$ distinct elements:
     $$G = \{e, x, y, xy\}.$$

3. This group presentation corresponds to the Klein four-group ($V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$), which has order 4.

Therefore, the maximum number of elements in such a group is 4.