GATE CS 2022 Set 1 — Question 27
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Engineering Mathematics → Sets & Combinatorics → Groups, Rings & Fields
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Question
Which of the following statements is/are TRUE for a group ?
Correct answer
(A) If for all x, y ∈ G, (xy)² = x² y², then G is commutative.; (B) If for all x ∈ G, x² = 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.
Solution
1.Statement (A): Given . Expanding, . By left cancellation of and right cancellation of , we get . Thus, is commutative (Abelian). True.
2.Statement (B): Given for all . This implies . For any , consider . Since , . Since elements are their own inverses, . Thus . True.
3.Statement (C): If the order of is 2, let . The only possible operation table forces and . This is cyclic and commutative. Generally, any group of prime order is cyclic and thus commutative. True.
4.Statement (D): If a group is commutative, all its elements commute. Therefore, elements in any subgroup must also commute. A subgroup of an Abelian group is always Abelian. False.
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