Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
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 .
is the set of bijective homomorphisms (Group isomorphisms, automorphisms and the set ).
The inverse of a bijective homomorphism is a homomorphism (The inverse of a bijective group homomorphism is a group homomorphism).
The symmetric group uses composition of bijections (The symmetric group : the bijections of a set under composition).
Bijections of a set form a group under composition ( is a group under composition, and it is non-abelian whenever has at least three distinct elements).
Proof
The identity map is an automorphism, and the composite of two automorphisms is again a bijective homomorphism.
By [L2], the inverse of every automorphism is an automorphism, while associativity comes from composition of functions.
Hence the closure and inverse properties in step 2.1 give a group structure on .
Depends on
- Group isomorphisms, automorphisms and the set $\operatorname{Aut}(G)$
- The inverse of a bijective group homomorphism is a group homomorphism
- The symmetric group $\operatorname{Sym}(X)$: the bijections of a set $X$ under composition
- $\operatorname{Sym}(X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements
Used by
- There are exactly two isomorphism classes of groups of order 105 Corollary
- Almost simple finite groups Definition
- Inner automorphisms and Inn(G) Definition
- The holomorph Hol(G)=G rtimesAut(G) Definition
- Automorphisms act linearly on the Frattini quotient Theorem
- 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 · two levels
12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Milne, Group Theory, Automorphisms of Groups (standard reference, not scraped)