GATE 2022 CS Q.27: solution and answer

GATE 2022 · CSQ.27 · 1 mark · MSQ

Which of the following statements is/are TRUE for a group $G$ ?

(A)
If for all $x, y \in G$ , $(x y)^{2} = x^{2} y^{2}$ , then $G$ is commutative.
(B)
If for all $x \in G$ , $x^{2} = 1$ , then $G$ is commutative. Here, $1$ is the identity element of $G$ .
(C)
If the order of $G$ is $2$ , then $G$ is commutative.
(D)
If $G$ is commutative, then a subgroup of $G$ need not be commutative.

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The true statements are A, B and C. D is false.

Official answer: A, B, CSource: Official answer key, GATE 2022 CS (IIT Kharagpur)

(A) Assume $(x y)^{2} = x^{2} y^{2}$ for all $x, y \in G$ . Then $x y x y = xx y y$ . Multiply on the left by $x^{- 1}$ and on the right by $y^{- 1}$ to get $y x = x y$ . So $G$ is commutative. TRUE.
(B) If $x^{2} = 1$ for all $x$ , every element is its own inverse, so $x^{- 1} = x$ . Then $x y = (x y)^{- 1} = y^{- 1} x^{- 1} = y x$ for all $x, y$ . So $G$ is commutative. TRUE.
(C) A group of order $2$ is ${1, a}$ . Every element is a power of $a$ , so the group is cyclic, and a cyclic group is commutative. TRUE.
(D) Let $H$ be a subgroup of a commutative group $G$ . For $h_{1}, h_{2} \in H \subseteq G$ we have $h_{1} h_{2} = h_{2} h_{1}$ , because the operation is the same as in $G$ . So every subgroup of a commutative group is commutative. The statement says a subgroup need not be commutative, so it is FALSE.
Correct options: A, B and C.

KEY IDEA

Commutativity comes from cancelling in $x y x y = xx y y$ (A), from $x = x^{- 1}$ giving $x y = (x y)^{- 1} = y x$ (B), and from a group of order $2$ being cyclic (C). A subgroup inherits commutativity from its parent group, so D fails.

WATCH OUT FOR

Calling D true by confusing it with the converse. A subgroup of a commutative group is always commutative. It is a non-commutative group that can have commutative subgroups.

From the GATE 2022 CS paper. Every question of this paper · Download the full paper · All 7 Groups questions, with notes

GATE 2022 CS Q.27: solution and answer

Answer and solution

More Groups questions