[[defining ingredient::Inner automorphism]]

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	Quick phrase	Defining ingredient
Central factor	factor in central product product with centralizer is whole group quotient action by outer automorphisms is trivial every inner automorphism restricts to an inner automorphism	Inner automorphism Centralizer Image-potentially operator Direct factor Upper join-closure operator Cocentral subgroup
Conjugacy class		Orbit Inner automorphism Conjugate elements
Dedekind group	every subgroup is normal every cyclic subgroup is normal normal closure of element is cyclic	Normal subgroup Inner automorphism Power automorphism Normal closure Levi operator Cyclic group Hamiltonian operator
Group in which every automorphism is inner		Normal subgroup Central factor Automorphism of a group Inner automorphism
Group in which every class-preserving automorphism is inner		Class-preserving automorphism Inner automorphism Conjugacy-closed normal subgroup Central factor
Group in which every normal-extensible automorphism is inner		Normal-extensible automorphism Inner automorphism
Group whose inner automorphism group is central in automorphism group		Inner automorphism group Central subgroup Automorphism group of a group Inner automorphism Central automorphism Normal subgroup Commutator-in-center subgroup
Inner-centralizing automorphism		Inner automorphism
Normal subgroup	invariant under inner automorphisms, self-conjugate subgroup same left and right cosets kernel of a homomorphism subgroup that is a union of conjugacy classes	Homomorphism of groups Homomorphism of groups Inner automorphism Conjugate subgroups Left coset of a subgroup Right coset of a subgroup Conjugacy class Commutator of two subgroups
SCAB-subgroup		Inner automorphism Subgroup-conjugating automorphism
Second center		Center Inner automorphism Upper central series
Subgroup with canonical abelianization		Inner automorphism IA-automorphism In-normalizer operator IA-balanced subgroup

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