Question

(3.6) Let H be a subgroup of G. Show that CG​(H) is a normal subgroup of NG​(H).

Answer 1

To show that CG(H) is a normal subgroup of NG(H), we need to prove that CG(H) is closed under conjugation by any element of NG(H).

Step-by-step explanation:

In order to show that CG(H) is a normal subgroup of NG(H), we need to prove that CG(H) is closed under conjugation by any element of NG(H).

Let g be an arbitrary element of NG(H). We want to show that for any h in CG(H), ghg^(-1) is also an element of CG(H).

Since h is in CG(H), we know that for any x in H, hx = xh. Therefore, for any x in H, we have ghxg^(-1) = gxg^(-1)h, which implies that ghg^(-1) is in CG(H).