Hint: z commutes with all elements of G.
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?
no, Z(G) can be lots bigger than just {e}. in fact, if G is abelian Z(G) is all of G!
however, for any z in Z(G), we have zx = xz for any element x of G. and this means that:
for any x in G, that is, is the identity automorphism of G.
you'll be done if you can prove that for any g,h in G (why?)