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.
A group homomorphism is injective if and only if its kernel is trivial
Statement
A group homomorphism is injective if and only if its kernel is trivial.
For a group homomorphism , is injective exactly when .
Facts & Assumptions
Given: A group homomorphism .
Homomorphisms preserve inverses and products (A group homomorphism automatically satisfies and , and for every ; for monoid homomorphisms preservation of the identity must be assumed).
Injectivity means that equal values have equal arguments (Injection, surjection, bijection).
In a group, implies (In a group , and , the order of the last product being essential).
Proof
If is injective and , then , so .
Conversely, if and , then , whence and .
The two implications prove the equivalence.