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 center of a group
Definition
Let be a group (Group and abelian group). The center of is
Thus consists of the elements that commute with every element of . Its subgroup and normality properties are proved in The center of a group is a normal subgroup.
Depends on
Used by
- G/Z(G)congInn(G) Corollary
- The center of a free product with at least two nontrivial factors is trivial Corollary
- S₃ has trivial center, so the finite p-group hypothesis in the nontrivial-center theorem is necessary Counterexample
- Every inner automorphism of an abelian group is the identity Example
- For an abelian group G, Z(G)=G and [G,G]={e} Example
- If G/Z(G) is cyclic, then G is abelian Lemma
- The center of a group is a normal subgroup Lemma
- Every nontrivial finite p-group has nontrivial center, in fact p divides |Z(P)| Theorem
- Every nontrivial normal subgroup of a finite p-group meets the center nontrivially Theorem
- The class equation |G|=|Z(G)|+∑ᵢ [G:C_G(xᵢ)] for a finite group Theorem
- The map g↦(x↦ gxg⁻¹) is a homomorphism GtoAut(G) with kernel Z(G) and image Inn(G) Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 9 results over 9 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, Characteristic subgroup (standard reference, not scraped)