[[Fact about::Characteristic subgroup]]

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	Uses	Fact about
AEP upper-hook characteristic implies AEP		AEP-subgroup Characteristic subgroup
Additive group of a field implies characteristic in holomorph	Additive group of a field implies monolith in holomorph Monolith is characteristic	Additive group of a field Holomorph-characteristic group Characteristically simple group Elementary abelian group Characteristic subgroup Holomorph of a group
Analogue of critical subgroup theorem for finite solvable groups		Finite solvable group Characteristic subgroup Frattini-in-center group Self-centralizing subgroup Coprime automorphism-faithful subgroup
Center is characteristic		Center Characteristic subgroup
Centerless and characteristic in automorphism group implies automorphism group is complete	Centerless implies inner automorphism group is centralizer-free in automorphism group Inner automorphism group is normal in automorphism group Normal and centralizer-free implies automorphism-faithful Automorphism group action lemma	Centerless group Automorphism group of a group Characteristic subgroup Complete group
Characteristic and self-centralizing implies coprime automorphism-faithful		Self-centralizing characteristic subgroup Coprime automorphism-faithful subgroup Characteristic subgroup Self-centralizing subgroup Coprime automorphism-faithful characteristic subgroup
Characteristic central factor of WNSCDIN implies WNSCDIN	Characteristic of normal implies normal	Composition operator Characteristic central factor WNSCDIN-subgroup Left-transitively WNSCDIN-subgroup Characteristic subgroup Central factor
Characteristic direct factor not implies fully invariant		Characteristic direct factor Fully invariant subgroup Characteristic subgroup Direct factor
Characteristic equals fully invariant in odd-order abelian group		Odd-order abelian group Characteristic subgroup Fully invariant subgroup Characteristic subgroup of odd-order abelian group Fully invariant subgroup of odd-order abelian group
Characteristic equals strictly characteristic in Hopfian		Hopfian group Characteristic subgroup Strictly characteristic subgroup Characteristic subgroup of Hopfian group Strictly characteristic subgroup of Hopfian group
Characteristic equals verbal in free abelian group		Free Abelian group Characteristic subgroup Verbal subgroup Characteristic subgroup of free Abelian group Verbal subgroup of free Abelian group
Characteristic implies automorph-conjugate		Characteristic subgroup Automorph-conjugate subgroup
Characteristic implies normal	Group acts as automorphisms by conjugation	Characteristic subgroup Normal subgroup
Characteristic not implies amalgam-characteristic		Characteristic subgroup Amalgam-characteristic subgroup
Characteristic not implies characteristic-isomorph-free		Characteristic subgroup Characteristic-isomorph-free subgroup
Characteristic not implies direct factor		Characteristic subgroup Direct factor
Characteristic not implies elementarily characteristic		Characteristic subgroup Elementarily characteristic subgroup
Characteristic not implies fully invariant		Characteristic subgroup Fully invariant subgroup
Characteristic not implies fully invariant in class three maximal class p-group		Maximal class group Characteristic subgroup Fully invariant subgroup
Characteristic not implies fully invariant in finite abelian group		Finite abelian group Characteristic subgroup Fully invariant subgroup

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