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

Depends on

• Group and abelian group
• Subgroup

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
• Double cosets K backslash G/H of two subgroups Definition
• Normal subgroup: invariance under conjugation Definition
• The Bass-Serre tree of a graph of groups Definition
• The labeled Schreier coset graph of a subgroup of a free group Definition
• The quotient group G/N and coset product (gN)(hN)=ghN Definition
• The transversal data used for HNN normal forms 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
• Frobenius reciprocity for group representations without tensor products Example
• G/{e} reproduces G, while G/G is the one-element quotient group Example
• The permutation representation on the left cosets G/H 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
• A left transversal identifies Ind_H^G W with a direct sum of [G:H] copies of W Proposition
• A subgroup of finite index in a finitely generated group is finitely generated, and its inclusion is a quasi-isometry Proposition
• Every conjugacy class of an extraspecial p-group outside the centre has exactly p elements Proposition
• The canonical projection π:G→ G/N, π(g)=gN, is a surjective group homomorphism Proposition
• For a nonempty path-connected total space, a covering fibre is in bijection with the right cosets of the induced fundamental-group subgroup Theorem
• Inducing the trivial representation gives the permutation representation on G/H Theorem
• 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 · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)