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.
The kernel and image of a group homomorphism
Definition
The kernel and image of a group homomorphism.
For a group homomorphism , define
Thus is surjective exactly when (Injection, surjection, bijection).
Depends on
Used by
- A subgroup is normal if and only if it is the kernel of a group homomorphism Corollary
- The alternating group Aₙ=ker(sgn) of even permutations Definition
- For n≥2, reduction ℤ→ℤ/n has kernel nℤ and realises ℤ/n by the first isomorphism theorem Example
- The trivial homomorphism G→ H has kernel G and image {e_H} Example
- Two elements have the same image under a homomorphism if and only if they lie in the same coset of its kernel Lemma
- The canonical projection R→ R/I is a surjective ring homomorphism with kernel I Proposition
- A group homomorphism is injective if and only if its kernel is trivial Theorem
- A homomorphism that kills a normal subgroup factors uniquely through the quotient group Theorem
- Correspondence theorem: subgroups of G/N correspond to subgroups of G containing N, with normality preserved Theorem
- Every group admits a presentation Theorem
- Left multiplication on G/H is transitive, has stabiliser H at H, and has kernel Core_G(H) Theorem
- The image of a group homomorphism is a subgroup and its kernel is a normal subgroup Theorem
- The kernel of a ring homomorphism is a two-sided ideal Theorem
- The map g↦(x↦ gxg⁻¹) is a homomorphism GtoAut(G) with kernel Z(G) and image Inn(G) Theorem
- Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group 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
- Ernst, An Inquiry-Based Approach to Abstract Algebra, Homomorphisms (standard reference, not scraped)