Question

3.5) Let H be a subgroup of G, and recall that CG​(H):={g∈G∣∀h∈H:gh=hg}. Show that CG​(Hg)=CG​(H)g for each element g in G.

Answer 1

To show that CG(Hg) = CG(H)g for each element g in G, we need to prove two conditions.

Step-by-step explanation:

To show that CG(Hg) = CG(H)g for each element g in G, we need to prove that for every g in G:

1. If ghg^(-1) = hg^(-1) for all h in H, then ghg^(-1)g = hg^(-1)g for all h in H.
2. If ghg^(-1)g = hg^(-1)g for all h in H, then gh = hg for all h in H.
3. Therefore, CG(Hg) = CG(H)g for each element g in G.