Showing that a subset is closed under a binary operation

[Mr Davis 97]

Homework Statement

Suppose that * is an associative binary operation on a set S. Let $H=\{a \in S ~| ~a*x=x*a, ~ \forall x \in S\}$. Show that * is closed under H.

Homework Equations

The Attempt at a Solution

Let b and c be two different elements in H. We need to show that b*c is also in H.

We know that bx = xb, and that cx = xc. Putting these two equations, and using associativity and commutativity the farthest I can get is (bc)xx = xx(bc). I'm not sure how to get (bc)x = x(bc)

 

[fresh_42]

Homework Statement

Suppose that * is an associative binary operation on a set S. Let $H=\{a \in S ~| ~a*x=x*a, ~ \forall x \in S\}$. Show that * is closed under H.

Homework Equations

The Attempt at a Solution

Let b and c be two different elements in H. We need to show that b*c is also in H.

We know that bx = xb, and that cx = xc. Putting these two equations, and using associativity and commutativity the farthest I can get is (bc)xx = xx(bc). I'm not sure how to get (bc)x = x(bc)

You start with $(bc)x$. And it's " H is closed under * ".