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
• 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
• 1→⟨ i⟩→ Q₈→ Q₈/⟨ i⟩→1 does not split, with nonabelian middle group Counterexample
• An HNN extension with its stable letter Definition
• Group presentation by generators and relations 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 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
• 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
• Central factors are equivalent to adjacent commutator containments Lemma
• Elementary groups are supersolvable Lemma
• Every finite abelian group is a quotient of (ℤ/n)ᵏ for some n and k Lemma
• If G/Z(G) is cyclic, then G is abelian Lemma
• If K is normal in G, N is normal in G and K⊆ N, then N/K is normal in G/K Lemma
• Monomiality lifts along a quotient Lemma
• The commutator pairing is well defined on the central quotient, is bilinear over Fₚ, and is alternating Lemma
• The commutator pairing of an extraspecial p-group has trivial radical Lemma
• The kernels of the finite coordinate projections form an open normal neighbourhood basis at the identity Lemma
• The square map is well defined on the central quotient and satisfies q(x̄ȳ)=q(x̄)+q(ȳ)+b(x̄,ȳ) Lemma
• The successive quotients pⁱG/pⁱ⁺¹G recover the cyclic summand multiplicities of a finite abelian p-group Lemma
• The transitive subgroups of S₄ and their action on the three pairings Lemma
• Two elements of an extraspecial p-group with nontrivial commutator generate an extraspecial subgroup of order p³ Lemma
• Abelian groups and ℤ-modules have the same objects and morphisms Proposition
• Dih(C₄) and Q₈ are extraspecial of order 8, with six and two solutions of x²=1 respectively Proposition
• Every intermediate field of ℚ(μₙ)/ℚ is Galois over ℚ with abelian Galois group Proposition
• In ⟨ X∣ R⟩, the words u and v represent the same element if and only if u⁻¹v∈⟨⟨ R⟩⟩ Proposition

…and 28 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)