# Normal equals potentially characteristic

From Groupprops

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

## Contents

## Statement

The following are equivalent for a subgroup of a group :

- 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 .

## Related facts

### Stronger facts

- 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

## Facts used

- Characteristicity is centralizer-closed
- Characteristic implies normal
- Normality satisfies intermediate subgroup condition

## Proof

### 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 .

**Proof**:

- 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 .

**Proof**:

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. |