The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$

Depends on

• Normal subgroup: invariance under conjugation
• The coset set $G/H$ and the index $[G:H]$ of a subgroup
• Left and right cosets $gH$ and $Hg$ of a subgroup

Used by

• An extraspecial p-group has order p¹⁺²ⁿ for some n≥1 Corollary
• An extraspecial p-group is the product of two maximal abelian subgroups meeting in its centre Corollary
• Converse of Lagrange for finite abelian groups: every divisor occurs as a subgroup order Corollary
• Frobenius automizer criterion for p nilpotence Corollary
• If one set in a van Kampen cover is simply connected, the other fundamental group surjects with overlap-generated kernel Corollary
• The centre of an extraspecial p-group has no complement Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• 1→⟨ i⟩→ Q₈→ Q₈/⟨ i⟩→1 does not split, with nonabelian middle group Counterexample
• Cyclic sylow does not alone imply a normal p complement Counterexample
• An HNN extension with its stable letter Definition
• Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order Definition
• Artin monoid and Artin group presentations, and the canonical monoid-to-group map Definition
• Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups Definition
• Group presentation by generators and relations Definition
• Harish-Chandra induction and restriction for finite general linear groups Definition
• P residual of a finite group Definition
• Permutation Weyl group and inversion length Definition
• Principal weil divisor and class group Definition
• Quotient module M/N with scalar multiplication on additive cosets Definition
• Supersolvable groups and monomial characters Definition
• The abelianisation Gᵃᵇ:=G/[G,G] and its canonical map Definition
• The central product G∘_α H of two groups along an isomorphism of central subgroups Definition
• The commutator pairing of an extraspecial p-group relative to a chosen generator of its centre Definition
• The Jacobian of a compact Riemann surface Definition
• The modular group and its action on the upper half-plane Definition
• The outer automorphism group Out(G)=Aut(G)/Inn(G) Definition
• The profinite completion is the inverse limit of the finite quotients G over N Definition
• The quotient ring R/I with (r+I)(s+I)=rs+I Definition
• The square map of an extraspecial 2-group relative to a chosen generator of its centre Definition
• The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums Definition
• The upper central series Definition
• Annihilators of closed subgroups of Euclidean space Example
• Central extensions of a cyclic group Example
• A compactly generated LCA group with no open subgroup of infinite index is Euclidean times compact Lemma
• A nonabelian supersolvable group has a noncentral abelian normal subgroup, also in its nonabelian quotients Lemma
• A nonabelian supersolvable group has a noncentral normal abelian subgroup Lemma
• A nontrivial finite abelian p-group with a unique subgroup of order p is cyclic Lemma
• A product formula for the number of square roots of the identity in a central product of extraspecial 2-groups Lemma
• A subgroup of the central quotient and its orthogonal complement have orders multiplying to the order of the quotient Lemma
• Annihilators reverse inclusions and the double annihilator closes the subgroup Lemma

…and 70 more results.

Dependency tree · two levels

8 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

• T. W. Judson, Abstract Algebra: Theory and Applications, Factor Groups and Normal Subgroups (standard reference, not scraped)