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.
Normal subgroup: invariance under conjugation
Definition
Let be a group and let be a subgroup (Subgroup). For , write
The subgroup is normal in when
In that case write . Equivalently, every inner conjugation of maps onto itself. The connection with equality of the left and right cosets of Left and right cosets and of a subgroup is proved in Equivalent characterisations of a normal subgroup by conjugates and left and right cosets.
Depends on
Used by
- A subgroup is normal if and only if it is the kernel of a group homomorphism Corollary
- Every subgroup of index p in a finite p-group is normal Corollary
- Internal direct products of finitely many normal subgroups Definition
- The normal closure of a subset of a group Definition
- The quotient group G/N and coset product (gN)(hN)=ghN Definition
- Core_G(H) is the largest normal subgroup of G contained in H Lemma
- If H≤ G and N is normal in G, then HN is a subgroup and H∩ N is normal in H Lemma
- If K is normal in G, N is normal in G and K⊆ N, then N/K is normal in G/K Lemma
- The normal closure of R is the set of finite products of conjugates of elements of R and their inverses Proposition
- Equivalent characterisations of a normal subgroup by conjugates and left and right cosets Theorem
- Every nontrivial normal subgroup of a finite p-group meets the center nontrivially Theorem
- The image of a group homomorphism is a subgroup and its kernel is a normal subgroup Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 10 results over 10 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
- Encyclopedia of Mathematics, Normal subgroup (standard reference, not scraped)