Which of the following statements is/are TRUE for a group G?(A) If for all x, y in G, (xy)^2 = x^2 *

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GATE-CS is a competitive examination conducted in India. This question is from the 2022 GATE-CS Discrete Mathematics paper.

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Question

Which of the following statements is/are TRUE for a group G?(A) If for all x, y in G, (xy)^2 = x^2 * y^2, then G is commutative.(B) If for all x in G, x^2 = 1, then G is commutative (1 is the identity).(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.

Accepted Answer

Correct Answer: (A), (B), and (C)Concept -- Group Theory (from Abstract Algebra by Dummit and Foote; Discrete Mathematics):(A) TRUE: (xy)^2 = xyxy. x^2 y^2 = xxyy. If xyxy = xxyy, then left-multiply by x^{-1}: yxy = xyy. Right-multiply by y^{-1}: yx = xy. So G is commutative (abelian).(B) TRUE: If x^2 = 1 for all x in G, then x = x^{-1} for every element (every element is its own inverse). For any x, y in G: (xy)^2 = 1 means xyxy = 1, so xy = (xy)^{-1} = y^{-1}x^{-1} = yx. Therefore G is abelian.(C) TRUE: A group of order 2 has exactly 2 elements: the identity e and one other element a. The Cayley table is completely determined: a*a = e, e*a = a*e = a. This is commutative. In fact, every group of prime order is cyclic and therefore abelian.(D) FALSE: If G is commutative (abelian), then every subgroup of G is also abelian. This is because a subgroup inherits the operation from G, and commutativity of multiplication is an inherited property. Every subset of an abelian group that forms a group under the same operation is necessarily abelian.

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