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 com...

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. 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.

Accepted Answer

A, B, C Statement A: (xy)^2 = x^2*y^2 expands to xyxy = xxyy, and cancelling x on the left and y on the right gives yx = xy, so G is abelian. Statement B: If every element equals its own inverse (order 2), then xy = (xy)^{-1} = y^{-1}x^{-1} = yx, so G is abelian. Statement C: A group of order 2 has only the identity and one other element; both must commute (the group is isomorphic to Z_2). Statement D is false because every subgroup of an abelian group is itself abelian.