Using Isomorphism Theorem to show

[missavvy]

Homework Statement

If G is a group and aϵG, then the inner automorphism θ_a: G --> G is defined by θ_a(g) = aga^-1. Let Inn(G) = group of inner automorphisms and Z(G) = the centre of G.
Use the Isomorphism theorem to show G/Z(G)≅Inn(G).

Homework Equations

The Attempt at a Solution

Firstly, the inner automorphism θ_a: G --> G defined by θ_a(g) = aga^-1 is a group homomorphism, with its kernel being the centre of G, denoted Z(G). And its image = Inn(G).
So since the theorem says Im(θ) ≅ G/ker(θ), then G/Z(G)≅Inn(G).

Would I have to somehow prove that Z(G) = ker(θ)? I don't really know where to begin.
Likewise with Im(θ) = Inn(G).

Please let me know if this attempt is sort of close to the actual solution.
Thankss!

 

[eok20]

You want to show that the assigment $$a\mapsto \theta_a$$ is a homomorphism from G to Aut(G) (the group of automorphisms of G). Clearly the image of this map is Inn(G). From what you wrote I think you were looking at an individual $$\theta_a$$ (which, btw, the kernel can be bigger than the center since there can be elements that commute with $$a$$ but not with everything, so the center will be a subgroup of the kernel of an individual $$\theta_a$$)