Never mind. I figured it out. It follows since the center is a subgroup and has the property that x ∈Z(G), g∈G then gxg-1∈Z(G).
I'm just having an off night and while reading through an algebra book I came across that center Z(G) is the kernel in this, but I can't verify it to myself. Can anyone verify the reason for this?
This is what I'm talking about: Center (group theory) - Wikipedia, the free encyclopedia
Every element g of a group G, an inner automorphism defined by . Verfiy that is an automorphism of G for given . Now, let defined by .The kernel of this map, which maps to an identity automorphism, is .