Question

Let G be a group such that (xy)2 = xy, Ɐx, y ∈ G, then which of the following is true​

Answer 1

Answer:

xy2=y2x.

Step-by-step explanation:

(xy)−1(xy)2(yx)−1=(xy)−1(yx)2(yx)−1

=>(xy)(yx)−1=(xy)−1(yx)

then somehow we have

=>xy2=x((x−1y)x)2

=>xy2=x(x(yx−1))2

=>xy2=(xyx)−1(xyx−1)x

=>xy2=y2x

I'm positive that the solution above is wrong I just need confirmation.

This is my solution:

Since G is a group then for all x,y ∈ G there exists x−1 , y−1 ∈ G.

(xy)2=(yx)2

y−1x−1xyxy=y−1x−1yxyx

xy=y−1x−1yxyx

y=x−1y−1x−1yxyx

Take the left hand side of xy2=y2x and substitute y=x−1y−1x−1yxyx

xy2=x(x−1y−1x−1yxyx)2

xy2=x(x−1y−1x−1yxyx)(x−1y−1x−1yxyx)

xy2=xx−1y−1x−1yxyxx−1y−1x−1yxyx

xy2=y−1x−1yyxyx

xy2=y−1x−1y2xyx

xyxy2=y2xyx

y−1x−1xyxy2=y2xyxx−1y−1

xy2=y2x.

Please mark me as brainlist