Question

The center Z(G) of a group G is defined to be the set Z(G) = {x ∈ G|∀g ∈ G, gx = xg}. Prove that Z(G) ≤ G and in particular Z(G) ◅ G.

Answer 1

shouldn't be too bad. you need
closed
inverses are in there
what is your definition of normal?

Answer 2

if you use N is normal in G if \(\forall g\in G, g^{-1}ng\in N\) you get the right away

Answer 3

since if \(z\in Z(G), g \in G \) you have \(g^{-1}zg=zg^{-1}g=z\in Z(G)\)

Answer 4

you need close right? that is if \(z_1z_2\in Z(G) \implies z_1z_2\in Z(G)\) work directy from the definition, you know \(gz_1=z_1\) and \(gz_2=z_2g\) use that fact to show that
\[gz_1z_2=z_1z_2g\implies z_1z_2\in Z(G)\]

Answer 5

you got this? last thing you have to prove is that if \(z\in Z(G)\) then so is \(z^{-1}\)

Answer 6

yea i think i got it, thanks a ton

Answer 7

oh i see a typ above. i should have written :

if \(z_1\) and \(z_2\) are in Z, then \(z_1z_2\in Z\)

Answer 8

you also need in \(z\in Z\) then \(z^{-1}\in Z\) and for that you can use the fact that \(gz=zg\) and so \((gz)^{-1}=(zg)^{-1}\) i.e.
\[z^{-1}g^{-1}=g^{-1}z^{-1}\]