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Cartan-Brauer-Hua theorem, Normal implies join-transitively subnormal, Normal implies modular, Normal implies permutable, Normal of order equal to least prime divisor of group order implies central, Normality is commutator-closed, Normality is strongly intersection-closed, Normality satisfies image condition, Normality satisfies intermediate subgroup condition, Proper and normal in quasisimple implies central, Totally disconnected and normal in connected implies central
