If G is a group and aϵG, then the inner automorphism θa: G --> G is defined by θa(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).
The Attempt at a Solution
Firstly, the inner automorphism θa: G --> G defined by θa(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.