GATE CS 2014 Set 3 Q50 — Monoids and Groups

We are given a group $(G, *)$ generated by two elements $x$ and $y$. The relations satisfied by these elements are:

1. $x * x = e \implies x^2 = e \implies x^{-1} = x$
2. $y * y = e \implies y^2 = e \implies y^{-1} = y$
3. $x * y * x * y = e \implies (xy)^2 = e$
4. $y * x * y * x = e \implies (yx)^2 = e$

Let us simplify the relation $(xy)^2 = e$:

Multiply both sides by $y$ on the right:

Now, multiply both sides by $x$ on the right:

Since $xy = yx$, the generators $x$ and $y$ commute, meaning the group $G$ is abelian.

Every element in the group is a product of some number of $x$'s and $y$'s. Since $x$ and $y$ commute, and $x^2 = y^2 = e$, any element $g \in G$ can be simplified to the form:

Thus, the maximum possible distinct elements in the group are:

• $x^0 y^0 = e$
• $x^1 y^0 = x$
• $x^0 y^1 = y$
• $x^1 y^1 = xy$

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

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