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
• Frobenius automizer criterion for p nilpotence Corollary
• Frobenius semidirect product decomposition Corollary
• A solvable group that is not an M-group Counterexample
• Cyclic sylow does not alone imply a normal p complement Counterexample
• An internal semidirect product and a complement to a normal subgroup Definition
• Cuspidal representations and Harish-Chandra series Definition
• Free-presentation kernel data Definition
• Harish-Chandra induction and restriction for finite general linear groups Definition
• Inertia group and characters lying above a normal type Definition
• Internal direct products of finitely many normal subgroups Definition
• K₁ of a ring and the Whitehead group of a discrete group Definition
• Minimal normal subgroups and the socle of a finite group Definition
• Normal p complement and p nilpotent group Definition
• Ordered partitions and coordinate parabolics Definition
• P residual of a finite group Definition
• Permutation Weyl group and inversion length 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 modular group and its action on the upper half-plane Definition
• The normal closure of a subset of a group Definition
• The p-core Oₚ(G) as the largest normal p-subgroup Definition
• The p-prime core of a finite group 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
• Affine linear Frobenius groups over finite fields Example
• Frobenius normal two complement for S₃ Example
• S₃ as a Frobenius group Example
• The real affine group is amenable and nonunimodular Example
• 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 faithful irreducible with a noncentral abelian normal subgroup is induced from a proper inertia group Lemma

…and 63 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)