Question-Let G be a group and let g∈G. If z ∈ Z(G) (the center of a group G), show that the inner automorphism induced by g is the same as the inner automorphism induced by zg(that is ϕg = ϕzg) Hint: What is the definition of Z(G)?

Proof:

Since Z(G) just contains e, then z must equal e

so ϕg = ϕzg

ϕg=ϕeg

ϕg=ϕg

I'm sure I'm missing some stuff, but is this even close?