TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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The automorphisms of a group form a group under composition

Statement

The automorphisms of a group form a group under composition.

Facts & Assumptions

Given: A group $G$ .

[L1]

$\operatorname{Aut}(G)$ is the set of bijective homomorphisms $G\to G$ (Group isomorphisms, automorphisms and the set $\operatorname{Aut}(G)$ ).

[L2]

The inverse of a bijective homomorphism is a homomorphism (The inverse of a bijective group homomorphism is a group homomorphism).

[L3]

The symmetric group uses composition of bijections (The symmetric group $\operatorname{Sym}(X)$ : the bijections of a set $X$ under composition).

[L4]

Bijections of a set form a group under composition ( $\operatorname{Sym}(X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements).

Proof

technique · direct

1.1

The identity map is an automorphism, and the composite of two automorphisms is again a bijective homomorphism.

L1L2L3L4givenalgebra

2.1

By [L2], the inverse of every automorphism is an automorphism, while associativity comes from composition of functions.

step 1.1L1L2L3L4givenalgebra

3.1

Hence the closure and inverse properties in step 2.1 give a group structure on $\operatorname{Aut}(G)$ .

step 2.1∎

Depends on

Used by

Inner automorphisms and Inn(G) Definition
Inn(G) is a normal subgroup of Aut(G) Theorem
The map g↦(x↦ gxg⁻¹) is a homomorphism G toAut(G) with kernel Z(G) and image Inn(G) Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 28 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Milne, Group Theory, Automorphisms of Groups (standard reference, not scraped)