Conjugation by an element

Inner group automorphism

The inner group automorphism Inn⁡(𝐺) ⊴Aut⁡(𝐺) is a normal subgroup given by conjugation by an element, group i.e. if ˆ𝑔 ∈Inn⁡(𝐺) then ˆ𝑔(ℎ) =𝑔ℎ𝑔−1 for some 𝑔 ∈𝐺. It is hence the image of conjugation as a group action ̂⋅ :𝐺 →Aut⁡(𝐺).

Proof of normal subgroup

That Inn⁡(𝐺) is a subgroup follows from the fact that it is the image of the homomorphism ̂⋅ :𝐺 →Aut⁡(𝐺). Let 𝑔 ∈𝐺 and 𝜑 ∈Aut⁡(𝐺). Then for any ℎ ∈𝐺

(𝜑̂𝑔𝜑−1)(ℎ)=(𝜑̂𝑔)(𝜑−1(ℎ))=𝜑(𝑔𝜑−1(ℎ)𝑔−1)=𝜑(𝑔)ℎ𝜑(𝑔)−1=̂𝜑(𝑔)(ℎ)

thus 𝜑̂𝑔𝜑−1 =̂𝜑(𝑔) ∈Inn⁡(𝐺).

By the First isomorphism theorem, this is isomorphic to 𝐺/𝑍(𝐺), where the divisor is the centre.

---

tidy | en | SemBr