DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-01

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 coset set $G / H$ and the index $[G : H]$ of a subgroup

Definition

Let $H \leq G$ . The left coset set is

$G / H : = {g H : g \in G} .$

By The left cosets of a subgroup partition the group, its elements are exactly the blocks of the coset partition of $G$ . The index of $H$ in $G$ is

$[G : H] : = ∣ G / H ∣$

when $G / H$ is finite, with finite cardinality as in The cardinality $∣ A ∣$ of a finite set. If $G / H$ is not finite, write $[G : H] = \infty$ . Here $\infty$ is a symbol, not a natural number, and no arithmetic with it is defined.

The right coset set has the same finite or infinite size because Inversion induces a bijection $g H \mapsto H g^{- 1}$ from left cosets to right cosets gives an explicit bijection between the two coset sets. Thus the index does not depend on choosing left rather than right cosets.

The coset set $G / H$ and the index $[G : H]$ of a subgroup

Definition

Depends on

Dependency tree · 2 levels

Sources

The coset set G/HG/H and the index [G:H][G:H] of a subgroup

Definition

Depends on

Dependency tree · 2 levels

Sources

The coset set $G / H$ and the index $[G : H]$ of a subgroup