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GATE 2022 CS – Question 27

Engineering Mathematics · Discrete Mathematics: Monoids and Groups · 1 mark · Multiple select

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

  1. If for all $x,y\in G$, $(xy)^2=x^2y^2$, then $G$ is commutative.
  2. If for all $x\in G$, $x^2=1$, then $G$ is commutative. Here, 1 is the identity element of $G$.
  3. If the order of $G$ is 2, then $G$ is commutative.
  4. If $G$ is commutative, then a subgroup of $G$ need not be commutative.

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Show answer and explanation

Correct answer: (A) If for all $x,y\in G$, $(xy)^2=x^2y^2$, 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.

Explanation

A: $xyxy=xxyy$ gives $yx=xy$ by cancellation. B: $xy=(xy)^{-1}=y^{-1}x^{-1}=yx$. C: every group of order 2 is cyclic, hence abelian. D is false because a subgroup of an abelian group is abelian.