Let H be a subgroup of G The centralizer of H is defined to

Let H be a subgroup of G. The centralizer of H is defined to be C(H) = {x Element of G|xh = hx for all h Element of H}. Prove that C(H) is a subgroup of G.

Solution

First, 1 C(H) since 1h = h = h1 for all h G. To see that C(H) is closed
under inverses, let x C(H) and observe that for all h G,
xh = hx
x1(xh)x1 = x1(hx)x 1
(x1x)hx1 = x1h(xx1
hx1 = x1h.
Finally, to see that C(H) is closed under products let x, y C(H). Then, for all
h G,
(xy)h = x(yh) = x(hy) = (xh)y = (hx)y = h(xy),
so xy C(H).