# Characteristicity is strongly intersection-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., strongly intersection-closed subgroup property)

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## Statement

### Verbal statement

An arbitrary (possibly empty) intersection of characteristic subgroups of a group is a characteristic subgroup.

### Statement with symbols

Suppose is an indexing set and is a collection of characteristic subgroups of a group . Then, the intersection of subgroups is also a characteristic subgroup of .

## Related facts

### Generalizations

- Invariance implies strongly intersection-closed: Any invariance property (i.e., a property that can be expressed as invariance under a certain collection of functions) is strongly intersection-closed: an arbitrary intersection of subgroups with the property again has the property.

Other particular cases of this general result are:

- Normality is strongly intersection-closed
- Strict characteristicity is strongly intersection-closed
- Full characteristicity is strongly intersection-closed

### Analogues in other algebraic structures

- Characteristicity is strongly intersection-closed in Lie rings
- Derivation-invariance is strongly intersection-closed

### Related metaproperty satisfactions and dissatisfactions for characteristicity

- Characteristicity is transitive: A characteristic subgroup of a characteristic subgroup is characteristic.
- Characteristicity is strongly join-closed: The subgroup generated by a collection of characteristic subgroups is characteristic.
- Characteristicity is not finite-relative-intersection-closed
- Characteristicity does not satisfy intermediate subgroup condition
- Characteristicity does not satisfy transfer condition

## Definitions used

### Characteristic subgroup

`Further information: Characteristic subgroup`

## Proof

### Hands-on proof

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### Property-theoretic proof

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