Question

Prove that the centre of a group is always a normal subgroup

Answer 1

Answer:

ok

Step-by-step explanation:

As you said in a comment you already showed that it is normal. So I will only show that it is a subgroup.

Clearly it contains ee, since eg=geeg=ge.

Now, we will show that it is closed. Let a,b∈Ha,b∈H, we know that ∀g:ag=ga∀g:ag=ga and gb=gbgb=gb. Thus, gab=agb=abggab=agb=abg and thus ab∈Hab∈H.

Now we only have to show that every h∈Hh∈H has an inverse and we are done. Let h∈Hh∈H, we know that ∀g∈G:gh=hg∀g∈G:gh=hg, thus

h−1(gh)h−1h−1g(hh−1)h−1(ge)h−1g=h−1(hg