Let G be a group a Prove that G is Abelian if and only if xy

Let G be a group.

(a) Prove that G is Abelian if and only if (xy)^2 = (x^2)(y^2) for all x, y G.

(b) Prove that G is Abelian if and only if (xy)^2 = (y^2)(x^2) for all x, y G.

(c) Suppose x^2 = e (where e is the identity element in G) for all x G. Prove that G is Abelian.

Solution

Assume G is abelian .

ley x,y belong to G

then (xy)^2 = xyxy = xxyx

since G is abelian and of course xxyy = x^2y^2

thus (xy)^2 = x^2y^2

assume (xy)^2 = x^2y^2 for all x,y belong to G

=>

xyxy = xxyy

x^-1xyxy = x^-1xxyy

yxy = xyy

yxyy^-1 = xyyy^-2

yx = xy

Thus G is abelian