GATE CS 2022 Q17 — Monoids and Groups

Let us analyze each statement for a group $G$:

• Statement A: We are given $(xy)^2 = x^2 y^2$ for all $x, y \in G$.
  Expanding both sides: $$(xy)(xy) = (xx)(yy) \implies xyxy = xxyy$$ By applying the left cancellation law (multiplying by $x^{-1}$ on the left) and the right cancellation law (multiplying by $y^{-1}$ on the right), we get: $$yx = xy$$ Since this holds for all $x, y \in G$, $G$ is commutative. Thus, A is TRUE.
• Statement B: We are given $x^2 = 1$ for all $x \in G$, which implies $x = x^{-1}$.
  For any $x, y \in G$, their product $xy \in G$, so: $$xy = (xy)^{-1} = y^{-1} x^{-1}$$ Since $y^{-1} = y$ and $x^{-1} = x$, we have: $$xy = yx$$ Thus, $G$ is commutative. So, B is TRUE.
• Statement C: Any group of prime order is cyclic, and every cyclic group is commutative. Since $2$ is a prime number, any group of order $2$ is cyclic and therefore commutative. Thus, C is TRUE.
• Statement D: If $G$ is commutative, then for any subgroup $H \subseteq G$ and any $h_1, h_2 \in H$, we have $h_1, h_2 \in G$. Since $G$ is commutative, $h_1 h_2 = h_2 h_1$ must hold. Thus, every subgroup of a commutative group is always commutative. So, D is FALSE.