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
• Mackey little-group reduction for an abelian normal subgroup 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
• A nontransitive system with two orbits is not classified by one stabilizer Counterexample
• A transitive action need not be Frobenius Counterexample
• Double cosets K backslash G/H of two subgroups Definition
• Normal subgroup: invariance under conjugation Definition
• Pointwise nonlinearities, neighborhood pooling, subgroup sampling, and coset pooling Definition
• Spherical subsets, the nerve, the poset of spherical cosets, and the Davis realization Definition
• Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups Definition
• The Bass-Serre tree of a graph of groups Definition
• The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset 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
• Transfer homomorphism for a finite index subgroup Definition
• Transitive systems of imprimitivity and their normalized measure class Definition
• Transversal normal-form data for an amalgamated free product Definition
• [G:G]=1 and, for finite G, [G:{e}]=|G| Example
• Finite transitive G-sets recover the stabilizer-induction classification 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
• Little groups for the real ax+b group and its orientation-preserving subgroup Example
• Minimal coset representatives of S₂ in S₃ Example
• Poincare products for Sn, Bn and Dn by explicit insertion Example
• The permutation representation on the left cosets G/H Example
• The regular translation system on L²(ℝⁿ): position, momentum and trivial stabilizer Example
• The right-angled cube Davis complex and its boundary 2-sphere Example
• Every left coset of a subgroup is itself a subgroup False statement
• A coset basis makes a monomial representation monomial matrices Lemma
• An induced representation carries a canonical system of imprimitivity on G/H Lemma
• Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups Lemma
• Classical Poincare products for A, B, D and I2(m) from the permutation and signed-permutation models Lemma
• Equality, inclusion and intersection of spherical cosets, and the quotient poset Lemma
• Every left or right coset of H is equinumerous with H Lemma
• Exceptional parabolic-orbit length certificates for E6, E7, E8, F4, H3 and H4 Lemma
• Fusion control forces trivial Sylow intersection with the p residual Lemma

…and 30 more results.

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)