DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02
How 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.
Group isomorphisms, automorphisms and the set
Definition
Group isomorphisms, automorphisms and the set .
An isomorphism is a bijective group homomorphism (Monoid homomorphism and group homomorphism, Injection, surjection, bijection). When , it is an automorphism of . Write
Depends on
Used by
- A pushout along an isomorphism is isomorphic to the other factor Corollary
- Free products are unique up to a unique factor-compatible isomorphism Corollary
- The cyclic group ℤ/4 is not isomorphic to ℤ/2×ℤ/2 Counterexample
- A path between basepoints induces an isomorphism of fundamental groups Example
- A pushout along an isomorphism recovers the other group Example
- The opposite-group functor is naturally isomorphic to the identity functor by inversion Example
- The inverse of a bijective group homomorphism is a group homomorphism Lemma
- Cayley's theorem: every group G is isomorphic to a subgroup of Sym(G) Theorem
- Conjugation x↦ gxg⁻¹ is an automorphism Theorem
- Every cyclic group is isomorphic to (ℤ,+) or to (ℤ/n,+) for its finite order n≥1 Theorem
- First isomorphism theorem for groups: G/ker fcongimf Theorem
- Free groups on the same set are uniquely isomorphic compatibly with their generators Theorem
- The automorphisms of a group form a group under composition Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 18 results over 12 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
- Judson, Abstract Algebra: Theory and Applications, Isomorphism Theorems (standard reference, not scraped)