Proof

We prove that the centralizer of a set X is a subgroup. The other cases are left to the reader.

• Clearly, the centralizer contains the unit element e.
• Suppose g and h are both centralizing X. Then gx = xg and hx = xh for all x X. Hence
  (gh)x = g(hx) = g(xh) = (gx)h = (xg)h = x(gh)

  and we find that gh also centralizes X.

• Moreover, if we multiply gx = xg from the right and left with g^-1 then we get
  xg^-1 = g^-1x

  and we conclude that g^-1 also centralizes X.

Hence the centralizer of X is a subgroup of G.