# Characteristicity is not finite-relative-intersection-closed

From Groupprops

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

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## Contents

## Statement

### Verbal statement

It is possible to have a group with subgroups , such that , characteristic in and characteristic in , but *not* characteristic in .

## Related facts

### Other facts about finite-relative-intersection-closed

- Subnormality is finite-relative-intersection-closed
- Transitive and transfer condition implies finite-relative-intersection-closed

### Related metaproperty satisfactions and dissatisfactions for characteristicity

- Characteristicity is transitive
- Characteristicity is strongly intersection-closed
- Characteristicity does not satisfy intermediate subgroup condition
- Characteristicity does not satisfy transfer condition

## Example

### Example where

`Further information: symmetric group:S4, dihedral group:D8, subgroup structure of symmetric group:S4`

Suppose is the symmetric group on the set . Define:

and define , explicitly, is a -Sylow subgroup of isomorphic to the dihedral group:

.

Then:

- is characteristic in : In fact, it is the only normal subgroup of order in , and also equals the second commutator subgroup of .
- is characteristic in : In fact, is an isomorph-free subgroup of .
- is not characteristic in : The intersection is the two-element subgroup , which is not characteristic -- in fact, it is not even normal in , as it is not invariant under conjugation by .

### Example where

`Further information: symmetric group:S4, dihedral group:D8, subgroup structure of symmetric group:S4`

Suppose is the symmetric group on the set . Define:

and is a -Sylow subgroup of isomorphic to the dihedral group, containing :

.

Then:

- is characteristic in : In fact, it is the only normal subgroup of order in , and also equals the second commutator subgroup of .
- is characteristic in : In fact, is the center of .
- is not characteristic in : The intersection is the two-element subgroup , which is not characteristic -- in fact, it is not even normal in , as it is not invariant under conjugation by .