Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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]

Aut⁡(G) is the set of bijective homomorphisms G→G (Group isomorphisms, automorphisms and the set 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 Sym⁡(X): the bijections of a set X under composition).

[L4]

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

Proof

technique · direct
1.1L1L2L3L4givenalgebra

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

2.1step 1.1L1L2L3L4givenalgebra

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

3.1step 2.1∎

Hence the closure and inverse properties in step 2.1 give a group structure on Aut⁡(G).

Depends on

Dependency tree · 2 levels

Sources