Normal closure (group theory)

In group theory, the normal closure of a subsetis the smallest normal subgroup ofcontainingFormally, ifis a subset ofthe normal closureis the intersection of all normal subgroups ofThe normal closureis the smallest normal subgroup ofis a subset of every normal subgroup ofThe subgroupis generated by the setof all conjugates of elements ofTherefore one can also writeAny normal subgroup is equal to its normal closure.The conjugate closure of the empty setis the trivial subgroup.[2] A variety of other notations are used for the normal closure in the literature, includingDual to the concept of normal closure is that of normal interior or normal core, defined as the join of all normal subgroups contained in[3] For a groupand defining relatorsthe presentation notation means thatis the quotient groupis a free group onThis group theory-related article is a stub.You can help Wikipedia by expanding it.
Normal closure (field theory)Algebraic structureGroup theorySubgroupNormal subgroupGroup actionQuotient group(Semi-)direct productDirect sumFree productWreath productGroup homomorphismssimplefiniteinfinitecontinuousmultiplicativeadditivecyclicabeliandihedralnilpotentsolvableGlossary of group theoryList of group theory topicsFinite groupsCyclic groupSymmetric groupAlternating groupDihedral groupQuaternion groupCauchy's theoremLagrange's theoremSylow theoremsHall's theoremp-groupElementary abelian groupFrobenius groupSchur multiplierClassification of finite simple groupsLie typesporadicDiscrete groupsLatticesIntegersFree groupModular groupsArithmetic groupLatticeHyperbolic groupTopologicalLie groupsSolenoidCircleGeneral linearSpecial linearOrthogonalEuclideanSpecial orthogonalUnitarySpecial unitarySymplecticLorentzPoincaréConformalDiffeomorphismInfinite dimensional Lie groupAlgebraic groupsLinear algebraic groupReductive groupAbelian varietyElliptic curvesubsetgeneratedconjugatesempty settrivial subgroupnormal corepresentationrelatorsSpringer-VerlagLyndon, Roger C.Schupp, Paul E.