DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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.

Internal direct products of finitely many normal subgroups

Definition

Let $G$ be a group and let $N_{0}, \dots, N_{r - 1}$ be normal subgroups, where $r \in N$ . They form an internal direct product when they generate $G$ and, for each $i < r$ , $N_{i} \cap ⟨ N_{j} : j < r, j \neq i ⟩ = {e} .$ The empty family is an internal direct product of the trivial group. For two subgroups of an abelian group this says $G = H K$ and $H \cap K = {e}$ ; in additive notation one writes $G = H \oplus K$ . Normal subgroups and generated subgroups are those of Normal subgroup: invariance under conjugation and The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups, and the comparison product is The external direct product $G \times H$ with componentwise multiplication.

Depends on

Dependency tree · 2 levels

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