Let $(G,+)$ be a group.
Let $a$ belong to $G$.
Let $C(a)=\{ g \in G \mid g+a=a+g\}$.
Does $C(a)$ satisfy the associative property?
WARNING (1) : It is not valid to prove that $C(a)$ is a subgroup. Only that property is required.
WARNING(2) : The operator $+$ is not the addition operator. It is an unknown operator.
I don't know how to prove the associative property because I don't know how to remove the parentheses so that I can operate freely.
For example:
Let $x,y,z$ be in $C(a)$, then $$(x+y)+z+a=(x+y)+a+z=x+y+a+z=x+a+(y+z) =a+x+(y+z).$$ Since $g+a=a+g$, the statement is proved.
However, I have removed the parentheses directly rather than by some kind of deduction. I think this cannot be done, right?
Let $S$ be any nonempty subset of $G$. Use concatenation for $+$. For any $r,s,t\in S$, since $S\subseteq G$, we can view $r,s,t\in G$. But then
$$r(st)=(rs)t$$
because $G$ is a group.