Here is the problem:
Suppose that $*$ is an associative binary operation on a set $S$. Let
$$H:= \{a \in S\mid a * x = x * a \mbox{ for all }x\in S\}.$$
In other words, $H$ is consisting of all the elements of $S$ that commute with every element in $S$.
Show that $H$ is closed under $*$.
Best Answer
Suppose that $a,b \in H$. You must show that $a * b \in H$. Thus, you must show that for any $x \in S$, $(a * b) * x = x * (a * b)$.
Here are some hints to get you started:
(1) Since $*$ is an associative operation, $(a * b) * x = a * (b * x)$.
(2) Since $b \in H$, $b * x = x * b$, so $a * (b * x) = a * (x * b)$.
Can you finish the proof from here? You're going to use a series of steps that are each similar to either (1) or (2).