Which of the following statements is/are TRUE for a group G ?
2022
Which of the following statements is/are TRUE for a group G ?
- A.
If for all x, y ∈ G, (xy)2 = x2y2, 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.
- D.
If G is commutative, then a subgroup of G need not be commutative.
Attempted by 109 students.
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Correct answer: A, B, C
Final answer: The true statements are the ones that assert the squared-product identity holds for all x,y in G, that every element squares to the identity, and that the group has order 2.
If for all x,y in G, (xy)^2 = x^2 y^2: From xyxy = xxyy, multiply on the left by x^{-1} and on the right by y^{-1} to get yx = xy, so G is abelian.
If for all x in G, x^2 = 1: Each element equals its inverse, so (xy)^2 = 1 implies xy = (xy)^{-1} = y^{-1} x^{-1} = yx, giving commutativity.
If the order of G is 2: Such a group is {1,a} with a^2 = 1, so it is cyclic and therefore abelian.
The remaining statement claiming that a subgroup of an abelian group need not be abelian is false: any subgroup of an abelian group is abelian because the group operation remains commutative when restricted to the subgroup.