# Difference between revisions of "Normal equals potentially characteristic"

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| 4 || <math>V</math> is characteristic in <math>K</math>. || || || Steps (2), (3) || Under any automorphism of <math>K</math>, the image of <math>V</math> is a homomorphic image of <math>V</math> in <math>K</math>. Its projection to <math>K/V \cong G</math> is a homomorphic image of <math>V</math> in <math>G</math>, which is trivial by step (3), so the image of <math>V</math> in <math>K</math> must be in <math>V</math>. | | 4 || <math>V</math> is characteristic in <math>K</math>. || || || Steps (2), (3) || Under any automorphism of <math>K</math>, the image of <math>V</math> is a homomorphic image of <math>V</math> in <math>K</math>. Its projection to <math>K/V \cong G</math> is a homomorphic image of <math>V</math> in <math>G</math>, which is trivial by step (3), so the image of <math>V</math> in <math>K</math> must be in <math>V</math>. | ||

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− | | 5 || The centralizer of <math>V</math> in <math>VH</math> equals <math>H</math>. || || || Steps (1), (2) || By definition of the wreath product action, <math>H</math> centralizes <math>V</math>. Since <math>S</math> is centerless, <math>V</math> is also centerless. Thus, <math>C_{VH}(V)</math> contains <math>H</math> but has trivial intersection with <math>V</math>, forcing <math>C_{VH} = H</math>. | + | | 5 || The centralizer of <math>V</math> in <math>VH</math> equals <math>H</math>. || || || Steps (1), (2) || By definition of the wreath product action, <math>H</math> centralizes <math>V</math>. Since <math>S</math> is centerless, <math>V</math> is also centerless. Thus, <math>C_{VH}(V)</math> contains <math>H</math> but has trivial intersection with <math>V</math>, forcing <math>C_{VH} (V)= H</math>. |

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| 6 || The centralizer of <math>V</math> in <math>K</math> equals <math>H</math>. || || || Steps (2), (5) || Step (5) already shows that <math>C_{VH}(V) = H</math>, so it suffices to show that <math>C_K(V) \le VH</math>. To see this, note that, by the construction in step (2), any element of <math>K</math> outside <math>VH</math> permutes the direct factors of <math>V</math> as an element of <math>G</math> outside <math>H</math>. This action is nontrivial, so the action is nontrivial, and hence elements outside <math>VH</math> cannot centralize <math>V</math>. This forces <math>C_K(V) \le VH</math>, completing the proof. | | 6 || The centralizer of <math>V</math> in <math>K</math> equals <math>H</math>. || || || Steps (2), (5) || Step (5) already shows that <math>C_{VH}(V) = H</math>, so it suffices to show that <math>C_K(V) \le VH</math>. To see this, note that, by the construction in step (2), any element of <math>K</math> outside <math>VH</math> permutes the direct factors of <math>V</math> as an element of <math>G</math> outside <math>H</math>. This action is nontrivial, so the action is nontrivial, and hence elements outside <math>VH</math> cannot centralize <math>V</math>. This forces <math>C_K(V) \le VH</math>, completing the proof. |

## Revision as of 17:03, 10 July 2019

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**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | 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 . | |||

2 | 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 . | Step (1) | |||

3 | Any homomorphism from to is trivial. | Steps (1), (2) | 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, any homomorphism from to is trivial. | ||

4 | is characteristic in . | Steps (2), (3) | 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 by step (3), so the image of in must be in . | ||

5 | The centralizer of in equals . | Steps (1), (2) | By definition of the wreath product action, centralizes . Since is centerless, is also centerless. Thus, contains but has trivial intersection with , forcing . | ||

6 | The centralizer of in equals . | Steps (2), (5) | Step (5) already shows that , so it suffices to show that . To see this, note that, by the construction in step (2), any element of outside permutes the direct factors of as an element of outside . This action is nontrivial, so the action is nontrivial, and hence elements outside cannot centralize . This forces , completing the proof. | ||

7 | is characteristic in . | Fact (1) | Steps (4), (6) | Step-fact combination direct. |

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

### 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. |