DefinitionDefinition: Literature-sourcedProof: 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.

Left and right cosets $g H$ and $H g$ of a subgroup

Definition

Let $G$ be a group and let $H \leq G$ be a subgroup (Group and abelian group, Subgroup). For $g \in G$ , the left coset and right coset of $H$ represented by $g$ are

$g H : = {g h : h \in H}, H g : = {h g : h \in H} .$

The element $g$ is a representative of these cosets. The notation denotes subsets of $G$ ; it does not assert that either subset is a subgroup.

Definition references

Group and abelian group

Subgroup

Remarks

Because the identity $e$ belongs to $H$ , every representative belongs to its two cosets: $g = g e = e g \in g H \cap H g$ .
The identity cosets are $e H = H = H e$ . Left and right cosets can differ in a nonabelian group.

Left and right cosets $g H$ and $H g$ of a subgroup

Definition

Definition references

Remarks

Depends on

Dependency tree · 2 levels

Sources

Left and right cosets gHgH and HgHg of a subgroup

Definition

Definition references

Remarks

Depends on

Dependency tree · 2 levels

Sources

Left and right cosets $g H$ and $H g$ of a subgroup