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## 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!!!