# Semantic search

Central factor implies normal, Central implies normal, Characteristic implies normal, Commutator of a group and a subgroup implies normal, Commutator of a normal subgroup and a subset implies 2-subnormal, Comparable with all normal subgroups implies normal in nilpotent group, Direct factor implies normal, Index three implies normal or double coset index two, Intermediately automorph-conjugate of normal implies weakly pronormal, Intermediately isomorph-conjugate of normal implies pronormal, Nilpotent implies every maximal subgroup is normal, Nilpotent implies every normal subgroup is potentially characteristic, Normal and self-centralizing implies coprime automorphism-faithful, Normal and self-centralizing implies normality-large, Normal implies join-transitively subnormal, Normal implies permutable, Potentially verbal implies normal, Subgroup of index equal to least prime divisor of group order is normal, Subgroup of index two is normal, Sylow of normal implies pronormal