Normal equals retract-potentially characteristic
This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
View a complete list of pages giving proofs of equivalence of definitions
- is a normal subgroup of .
- is a retract-potentially characteristic subgroup of in the following sense: there exists a group containing as a retract such that is a characteristic subgroup of .
- NPC theorem
- Finite NPC theorem
- Finite NIPC theorem
- Fact about amalgam-characteristic subgroups: finite normal implies amalgam-characteristic, periodic normal implies amalgam-characteristic, central implies amalgam-characteristic
Given: A group , a normal subgroup of .
To prove: There exists a group containing as a retract such that is characteristic in .
- Let be a simple non-abelian group that is not isomorphic to any subgroup of : Note that such a group exists. For instance, we can take the finitary alternating group on any set of cardinality strictly bigger than that of .
- Let be the restricted wreath product of and , where acts via the regular action of and let be the restricted direct power . In other words, is the semidirect product of the restricted direct power and , acting via the regular group action of .
- Any homomorphism from to is trivial: By definition, is a restricted direct product of copies of . Since is simple and not isomorphic to any subgroup of , any homomorphism from to is trivial. Thus, for any homomorphism from to is trivial.
- is characteristic in : Under any automorphism of , the image of is a homomorphic image of in . Its projection to is a homomorphic image of in , which is trivial, so the image of in must be in .
- The centralizer of in equals : By definition, centralizes . Using the fact that is centerless and that inner automorphisms of cannot be equal to conjugation by elements in , we can show that it is precisely the center.
- is characteristic in : This follows from the previous two steps and fact (1).