# Center is 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., 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 a characteristic subgroup of it.

## Related facts

### Stronger facts: stronger subgroup properties satisfied

The center satisfies a number of stronger properties:

Property | Meaning | Proof of satisfaction by center | Proof that it's stronger than characteristicity |
---|---|---|---|

strictly characteristic subgroup | invariant under all surjective endomorphisms | Center is strictly characteristic | strictly characteristic implies characteristic |

quasiautomorphism-invariant subgroup | invariant under all quasiautomorphisms | Center is quasiautomorphism-invariant | quasiautomorphism-invariant implies characteristic |

purely definable subgroup | can be defined in first-order language using the pure theory of the group | Center is purely definable | purely definable implies characteristic |

elementarily characteristic subgroup | no other elementarily equivalently embedded subgroups | Center is elementarily characteristic | elementarily characteristic implies characteristic |

bound-word subgroup | described as set of solutions of a system of equations, other variables quantified | Center is bound-word | (via strictly characteristic) |

### Stronger subgroup properties not satisfied

Property | Meaning | Proof of dissatisfaction | Proof that it is stronger than characteristicity |
---|---|---|---|

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

normal-homomorph-containing subgroup | contains any homomorphic image that is normal in whole group | center not is normal-homomorph-containing | (via strictly characteristic) |

weakly normal-homomorph-containing subgroup | contains any homomorphic image in a map that sends normal subgroups to normal subgroups | center not is weakly normal-homomorph-containing | (via strictly characteristic) |

1-automorphism-invariant subgroup | invariant under all 1-automorphisms | center not is 1-automorphism-invariant | 1-automorphism-invariant implies characteristic |

### Weaker facts: weaker subgroup properties satisfied

### Analogues in other algebraic structures

- Center is characteristic for Lie rings
- Center is derivation-invariant (another analogue for Lie rings)