Normal equals 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 potentially characteristic subgroup of in the following sense: there exists a group containing such that is a characteristic subgroup of .
- 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
- Characteristicity is centralizer-closed
- Characteristic implies normal
- Normality satisfies intermediate subgroup condition
Proof of (1) implies (2) (hard direction)
Given: A group , a normal subgroup of .
To prove: There exists a group containing 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).
Proof of (2) implies (1) (easy direction)
Given: A group , a subgroup of , a group containing such that is characteristic in .
To prove: is normal in .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||is normal in .||Fact (2)||is characteristic in||--||Given-fact-combination direct.|
|2||is normal in .||Fact (3)||Step (1)||Given-step-fact combination direct.|