# Center is purely definable

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., purely definable 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 purely definable subgroup: it can be defined as a subset in the first-order theory of the pure group.

## Related facts

### Weaker facts: weaker subgroup properties satisfied

Property | Meaning | Proof that it is satisfied by the center |
---|---|---|

elementarily characteristic subgroup | no other elementarily equivalently embedded subgroups | center is elementarily characteristic |

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

normal subgroup | invariant under all inner automorphisms | center is normal |

### Similar subgroup-defining functions being purely definable

Subgroup-defining function | Meaning | Relation to center | Proof that it is purely definable |
---|---|---|---|

member of the finite upper central series | ascending series obtained starting from trivial group where each member's quotient by predecessor equals center of group's quotient by predecessor | center is first nontrivial member | finite upper central series member is purely definable |

## Proof

We provide here the formula that an element satisfies if and only if it is in the center:

This is a first-order description, so the center is purely definable.