Normal subgroup: invariance under conjugation

Depends on

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

Used by

• A subgroup is normal if and only if it is the kernel of a group homomorphism Corollary
• A Sylow p-subgroup is normal if and only if it is unique Corollary
• Every subgroup of index p in a finite p-group is normal Corollary
• For n≥5, the only proper nontrivial normal subgroup of Sₙ is Aₙ Corollary
• An internal semidirect product and a complement to a normal subgroup Definition
• Free-presentation kernel data Definition
• Inertia group and characters lying above a normal type Definition
• Internal direct products of finitely many normal subgroups Definition
• Minimal normal subgroups and the socle of a finite group Definition
• Simple groups Definition
• Subnormal and normal series, factors, refinements, and equivalence Definition
• Supersolvable groups and monomial characters Definition
• The central product G∘_α H of two groups along an isomorphism of central subgroups Definition
• The finite residual is the intersection of the finite-index normal subgroups, and a group is residually finite when that intersection is trivial Definition
• The normal closure of a subset of a group Definition
• The p-core Oₚ(G) as the largest normal p-subgroup Definition
• The profinite topology on a group uses finite-index normal subgroups as an identity-neighbourhood basis Definition
• The quotient group G/N and coset product (gN)(hN)=ghN Definition
• The two-circle wedge has both regular and nonregular connected three-sheeted coverings Example
• V₄={1,(12)(34),(13)(24),(14)(23)} is a proper nontrivial normal subgroup of A₄ Example
• FALSE: Aₙ is simple for every n≥4 False statement
• A conjugacy class in an index-two normal subgroup remains one class or splits into two equal classes, with a centralizer criterion Lemma
• A nonabelian supersolvable group has a noncentral normal abelian subgroup Lemma
• A nontrivial normal subgroup of a solvable group contains a nontrivial abelian subgroup normal in the whole group Lemma
• A normal subgroup of Aₙ containing one 3-cycle equals Aₙ for n≥5 Lemma
• Characteristic subgroups are normal, and characteristicity is transitive Lemma
• Core_G(H) is the largest normal subgroup of G contained in H Lemma
• Distinct minimal normal subgroups centralize one another Lemma
• Distinct normal Sylow subgroups centralize one another Lemma
• Every nontrivial normal subgroup of Aₙ contains a 3-cycle for n≥5 Lemma
• If H≤ G and N is normal in G, then HN is a subgroup and H∩ N is normal in H Lemma
• If K is characteristic in N and N is normal in G, then K is normal in G Lemma
• If K is normal in G, N is normal in G and K⊆ N, then N/K is normal in G/K Lemma
• The centralizer of a normal subgroup is normal Lemma
• The Hopf-formula quotient exists Lemma
• The identified subgroup used to form a central product is central, hence normal Lemma
• The kernels of the finite coordinate projections form an open normal neighbourhood basis at the identity Lemma
• The transitive subgroups of S₄ and their action on the three pairings Lemma
• A representation with kernel containing a normal subgroup factors through the quotient, and irreducibility is unchanged by inflation Proposition
• Every intermediate field of ℚ(μₙ)/ℚ is Galois over ℚ with abelian Galois group Proposition

…and 17 more results.

Dependency tree · two levels

6 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

• Encyclopedia of Mathematics, Normal subgroup (standard reference, not scraped)