Choose the correct statement for a group G :

1. AIf for all x, y ∈ G, (xy)² = x²y², then G is commutative.
2. BIf for all x ∈ G, x³ = 1, then G is commutative (1 is the identity of G).
3. CIf for all x ∈ G, x⁵ = 1, then G is commutative (1 is the identity of G).
4. DIf G is commutative, a subgroup of G need not be commutative.

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Show answer & explanation

Correct answer: A. If for all x, y ∈ G, (xy)² = x²y², then G is commutative.

Explanation

(xy)² = x²y² means xyxy = xxyy; cancel x on the left and y on the right to get yx = xy, so G is abelian. Groups of exponent 3 or 5 can be non-abelian (e.g. the Heisenberg group mod 3), and every subgroup of an abelian group is abelian.