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Characteristically complemented characteristic is transitive, Cyclic characteristic implies hereditarily characteristic, Equivalence of normality and characteristicity conditions for isomorph-free p-functor, Finite NPC theorem, Intermediately characteristic of transfer-closed characteristic implies intermediately characteristic, Join of characteristic and characteristic-potentially characteristic implies characteristic-potentially characteristic, Kernel of a characteristic action on an abelian group with which it is characteristic in the direct product implies potentially characteristic, No nontrivial abelian normal p-subgroup for some prime p implies every p-divisible normal subgroup is potentially characteristic, Pi-separable and pi'-core-free implies pi-core is self-centralizing, Thompson's critical subgroup theorem, Transfer-closed characteristicity is transitive