 Centralizer of a subset

The centralizer of a subset $A$ of a group consists of the elements commuting with every member of $A$. It is the intersection of the element centralizers, hence a subgroup. Directly, identity commutes with every element, products of two elements that each commute with every member of $A$ still have that property, and the same is true of inverses. For $A=G$ this is the centre of a group. For an empty subset the commutation condition is vacuous and the centralizer is all of $G$.