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## Homework Statement

Show that the inner automorphisms of a group G form a normal subgroup of the group of all automorphisms of G under function composition.

## Homework Equations

in the previous problem, i showed that all automorphisms of a group G form a group under function composition

## The Attempt at a Solution

so i need help understanding the question. in the previous problem, assuming i did it correctly, i let A be the set of all automorphisms and showed it was a group in G. in this problem, am i to let say I be the set of all inner automorphisms and show it is a normal subgroup of A?

here is my previous proof.

: Claim: All automorphisms of a group G form a group under function composition.

Proof: Let A be the set of automorphisms of a group G, and let μ(g) and σ(g) be in A, with g in G. Since μ(g) and σ(g) are automorphisms, it follows that the mapping of μ(g)◦σ(g) defined as μ(σ(g)) is closed under function composition.

Let λ(g) be in A. Then

(μ(g)◦σ(g))◦λ(g) = μ(σ(g))◦λ(g)

= μ(g)◦σ(g)◦λ(g)

= μ(g)◦(σ(g)◦λ(g))

so A is associative.

Consider μ:G→G such that μ(g) = g. Then μ is the identity.

Consider σ:G→G such that σ(g) = a. Now consider λ:G→G such that λ(a) = g. Then σ(λ(a)) = σ(g) = a, and λ is the inverse of σ.

Since A is the set of automorphisms of G, it follows that for any σ in A, then σ is homomorphic. Hence, the set of automorphisms of a group G forms a group under function composition.