DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-01

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Left and right cosets $gH$ and $Hg$ of a subgroup

Definition

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

$gH:=\{gh:h\in H\},\qquad Hg:=\{hg:h\in H\}.$

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.

Remarks

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

Depends on

Used by

[G:H]=1 if and only if H=G Corollary
2ℤ has index 2 in ℤ and is nevertheless equinumerous with ℤ Counterexample
A left coset that is not the corresponding right coset in Sym({1,2,3}) Counterexample
A nonnormal two-element subgroup of Sym({1,2,3}) makes coset multiplication depend on representatives Counterexample
Normal subgroup: invariance under conjugation Definition
The quotient group G/N and coset product (gN)(hN)=ghN Definition
Transversal normal-form data for an amalgamated free product Definition
[G:G]=1 and, for finite G, [G:{e}]=|G| Example
For n≥1, the cosets of nℤ are the n congruence classes modulo n Example
G/{e} reproduces G, while G/G is the one-element quotient group Example
Every left coset of a subgroup is itself a subgroup False statement
Every left or right coset of H is equinumerous with H Lemma
Inversion induces a bijection gH↦ Hg⁻¹ from left cosets to right cosets Lemma
x∈ aH iff a⁻¹x∈ H, and aH=bH iff a⁻¹b∈ H Lemma
The canonical projection π:G→ G/N, π(g)=gN, is a surjective group homomorphism Proposition
Left multiplication on G/H is transitive, has stabiliser H at H, and has kernel Core_G(H) Theorem
Orbit-stabiliser: G/Gₓ→ G· x, gGₓ↦ g· x, is a well-defined bijection 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

Thomas W. Judson, Abstract Algebra: Theory and Applications, Cosets and Lagrange's Theorem (standard reference, not scraped)
Thomas W. Judson, Abstract Algebra: Theory and Applications, §6.1: Cosets (standard reference, not scraped)

Left and right cosets gHgHgH and HgHgHg of a subgroup