WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed
Sylow normalizer implies weakly abnormal
Central factor implies transitively normal
Normal Hall implies permutably complemented
Intermediately automorph-conjugate of normal implies weakly pronormal
Characteristic upper-hook AEP implies characteristic
Normal and self-centralizing implies coprime automorphism-faithful
Prime power order implies center is normality-large
Conjugate-join-closed subnormal implies join-transitively subnormal
Amalgam-characteristic implies potentially characteristic
Maximal among abelian normal implies self-centralizing in nilpotent
Subgroup of finite index has a left transversal that is also a right transversal
Subgroup of finite group has a left transversal that is also a right transversal
Commutator of a transitively normal subgroup and a subset implies normal
Left-transitively complemented normal implies characteristic
Base of a wreath product implies right-transitively conjugate-permutable
Pronormal implies self-conjugate-permutable
Central implies normal satisfying the subgroup-to-quotient powering-invariance implication
Central implies image-potentially characteristic
Amalgam-characteristic implies image-potentially characteristic
Finite normal implies amalgam-characteristic
Direct factor implies join-transitively central factor
Direct factor implies right-quotient-transitively central factor
Right-quotient-transitively central factor implies join-transitively central factor
Endomorphism kernel implies quotient-powering-invariant
Endomorphism image implies powering-invariant
Endomorphism image implies divisibility-closed
Divisibility-closed implies powering-invariant
Finite normal implies image-potentially characteristic
Central implies potentially characteristic
Normal and centralizer-free implies automorphism-faithful
3-subnormal implies finite-conjugate-join-closed subnormal
2-subnormal implies join-transitively subnormal
2-subnormal implies conjugate-permutable
Perfect subnormal implies join-transitively subnormal
Perfect subnormal implies subnormal-permutable
Self-centralizing and minimal normal implies strictly characteristic
Join of finitely many direct factors implies central factor
Characteristic central factor of WNSCDIN implies WNSCDIN
Pronormal implies intermediately subnormal-to-normal
Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal
Complemented normal implies endomorphism kernel
Join with any distinct conjugate is the whole group implies pronormal
Central implies potentially verbal in finite
Periodic normal implies potentially characteristic
Periodic normal implies amalgam-characteristic
Retract implies local divisibility-invariant
Nilpotent implies every maximal subgroup is normal
Fully invariant direct factor implies left-transitively homomorph-containing
Central subgroup implies join-transitively central factor
Cyclic-quotient characteristic implies upward-closed characteristic
Cyclic characteristic implies hereditarily characteristic
Normal and self-centralizing implies normality-large
Isomorph-containing implies characteristic
Commutator of a normal subgroup and a subset implies 2-subnormal
Normalizer of intermediately subnormal-to-normal implies self-normalizing
Left-transitively permutable implies characteristic
Normal of finite index implies quotient-powering-invariant
Cyclic normal implies hereditarily normal
Fully invariant of strictly characteristic implies strictly characteristic
Intersection of DPICF and direct factor is central factor
Retract implies central factor of normalizer
Subgroup of finite abelian group implies finite-abelian-pi-potentially verbal
Hall-extensible implies class-preserving
Finite normal implies quotient-powering-invariant
Commutator of a group and a subgroup implies normal
Potentially characteristic implies normal
Prehomomorph-contained implies strictly characteristic
Polynormal implies intermediately subnormal-to-normal
Identity functor controls strong fusion for abelian Sylow subgroup
Characteristic subgroup of abelian group implies powering-invariant
Base of a wreath product implies right-transitively 2-subnormal
Retract implies derived subgroup equals intersection with whole derived subgroup
Kernel of a characteristic action on an abelian group implies strongly image-potentially characteristic
Index four implies 2-subnormal or double coset index two
Marginal implies direct power-closed characteristic
Left-transitively 2-subnormal implies normal of characteristic
Normal-homomorph-containing implies strictly characteristic
Periodic normal implies image-potentially characteristic
Normality-preserving endomorphism-invariant implies finite direct power-closed characteristic
Normality-preserving endomorphism-invariant implies characteristic
Base of a wreath product implies subset-conjugacy-closed
SCDIN of subset-conjugacy-closed implies SCDIN
Finite index implies completely divisibility-closed
Nilpotent-quotient implies subgroup-to-quotient powering-invariance implication
Self-centralizing and minimal normal implies characteristic
Normal subgroup of nilpotent group satisfies the subgroup-to-quotient powering-invariance implication
Normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication
Intersection of kernels of bihomomorphisms implies completely divisibility-closed
Kernel of a bihomomorphism implies completely divisibility-closed
Characteristic and self-centralizing implies coprime automorphism-faithful
Characteristic subgroup of abelian group implies intermediately powering-invariant
2-subnormal implies conjugate-join-closed subnormal
Index three implies normal or double coset index two
Subhomomorph-containing implies right-transitively homomorph-containing
Conjugacy-closed implies focal subgroup equals derived subgroup
Intermediately normal-to-characteristic implies intermediately subnormal-to-normal
Finite solvable-extensible implies class-preserving
Normal implies join-transitively subnormal
Permutable and subnormal implies join-transitively subnormal
Normal-potentially characteristic implies normal-extensible automorphism-invariant
Free factor implies self-normalizing or trivial
Characteristic implies automorph-conjugate
Kernel of a bihomomorphism implies abelian-quotient
Cocentral implies right-quotient-transitively central factor
C-closed implies local powering-invariant
CEP implies every relatively normal subgroup is weakly closed
Normalizer of pronormal implies abnormal
Verbal subgroup of abelian group implies divisibility-closed
Cofactorial automorphism-invariant implies left-transitively 2-subnormal
Direct factor implies transitively normal
Normal of finite index implies completely divisibility-closed
Every normal subgroup satisfies the quotient-to-subgroup powering-invariance implication
Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant
Intermediately isomorph-conjugate of normal implies pronormal
Characteristic subgroup of abelian group is quotient-powering-invariant
Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup
Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups
Fully invariant implies finite direct power-closed characteristic