I am new at group theory, and I came across a question I would like help with
Suppose we have a set $S$ with the only elements $p,q,r$. Let $a$ and $b$ be two elements of $S$. Consider the following properties of $S$:
1) $aa=a$
2) $ab=ba$
3) $(ab)c=a(bc)$
4)$pa=a$ for every element $a$
Prove that there exists some element in $b \in S$ such that $bp=b, bq=b, br=b$.
Thank you! I am new in Group Theory so i was just looking for some help