Function restriction formalism chart

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Subgroup property Function restriction expression Endo-invariance Balanced Invariance Left-inner Quotient-hereditary invariance
Normal subgroup Inner \to Aut, Inner \to Function Yes No Yes Yes Yes
Characteristic subgroup Aut \to Aut, Aut \to Function Yes Yes Yes No No
Strictly characteristic subgroup Surj. End \to Function, Surj. End. \to End. Yes No Yes No Yes
Fully invariant subgroup End. \to End. Yes Yes Yes No Yes
Injective endomorphism-invariant subgroup Inj. End.\to End., Inj. end. \to Inj. End. Yes Yes Yes No No
Central factor Inner \to Inner No Yes No Yes No
Transitively normal subgroup Normal \to Normal, Inner \to Normal No Yes No Yes No
Conjugacy-closed normal subgroup Class \to Class, Inner \to Class No Yes No Yes  ?
Retraction-invariant subgroup Retraction \to Retraction, Retraction \to Function Yes Yes Yes No Yes

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