# Thread: proving * is commutative

1. ## proving * is commutative

This is the request of the exercise:

"Knowing (M,*) is a grupoid and the following statements are true:
1.x*x=x
2.(x*y)*z=(y*z)*x

Prove that * is commutative.

Thanks!

2. Originally Posted by cristidrincu
This is the request of the exercise:

"Knowing (M,*) is a grupoid and the following statements are true:
1.x*x=x
2.(x*y)*z=(y*z)*x

Prove that * is commutative.

Thanks!
What do you get if, in (2), you take z= x? Now "multiply", on the right, on both sides of that, by x.

3. ## proving * is commutative

Thanks. I'm new to abstract algebra. Did not know that we can substitute z with x. Thanks again!

4. Originally Posted by HallsofIvy

What do you get if, in (2), you take z= x? Now "multiply", on the right, on both sides of that, by x.
i don't see how this will solve the problem? here is my solution: $xy=(xy)(xy)=((xy)x)y=((yx)x)y=(x^2y)y=(xy)y=y^2x=y x.$