# Center is strictly characteristic

From Groupprops

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) always satisfies a particular subgroup property (i.e., strictly characteristic subgroup)}

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Contents

## Statement

The center of a group is always a strictly characteristic subgroup: any surjective endomorphism of the whole group sends the center to within itself.

## Related facts

### Stronger facts: stronger subgroup properties satisfied

Property | Meaning | Proof of satisfaction by the center | Proof that it is stronger than strict characteristicity |
---|---|---|---|

bound-word subgroup | described as a set of solutions to a system of equations, other variables quantified | center is bound-word | bound-word implies strictly characteristic |

### Stronger subgroup properties not satisfied

Property | Meaning | Proof of dissatisfaction by the center | Proof that it is stronger than strict characteristicity |
---|---|---|---|

normal-homomorph-containing subgroup | contains every homomorphic image that is normal in whole group | center not is normal-homomorph-containing | normal-homomorph-containing implies strictly characteristic |

fully invariant subgroup | invariant under all endomorphisms | center not is fully invariant | fully invariant implies strictly characteristic |

### Weaker facts: weaker subgroup properties satisfied

(Logically, the center satisfies all weaker properties, but we single out some important ones here).

Property | Meaning | Proof of satisfaction by the center | Proof that it is weaker than strict characteristicity |
---|---|---|---|

characteristic subgroup | invariant under all automorphisms | center is characteristic | strictly characteristic implies characteristic |

normal subgroup | invariant under all inner automorphisms | center is normal | (via characteristic) |

### Strict characteristicity for similar subgroup-defining functions

Subgroup-defining function (or collection of functions) | Meaning | Relation to center | Proof that it is strictly characteristic |
---|---|---|---|

member of upper central series | (transfinite) ascending series with each member's quotient by predecessor equal to center of group's quotient by predecessor | center is first member after trivial subgroup | upper central series members are strictly characteristic |

Baer norm | intersection of normalizers of all subgroups | contains center, contained in second member of upper central series | Baer norm is strictly characteristic |

Wielandt subgroup | intersection of normalizers of subnormal subgroup | contains center | Wielandt subgroup is strictly characteristic |

## Proof

**Given**: A group , with center . A surjective endomorphism .

**To prove**: .

**Proof**: Suppose . We need to show that for any , .

Since , there exists such that . Further, since is surjective, there exists such that . Since , we have:

.

Applying to both sides and using the property that is a homomorphism yields:

as required.