# Characteristicity is not upper join-closed

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

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## Contents

## Statement

### Property-theoretic statement

The subgroup property of being a characteristic subgroup does not satisfy the subgroup metaproperty of being upper join-closed.

### Statement with symbols

It is possible to find a group , a subgroup of , and intermediate subgroups of containing , such that is characteristic in both and , but is not characteristic in .

## Proof

### A generic example

Let be a group and be subgroups of such that:

- There is no nontrivial homomorphism from to
- There is no nontrivial homomorphism from to

Then, set . We then observe that:

- is a characteristic subgroup inside (in fact, is fully characteristic inside ).
- is a characteristic subgroup inside (in fact, is fully characteristic inside ).
- is
*not*characteristic inside .

Note that this generic example also shows the the property of being a fully characteristic subgroup is not closed under upper joins. `For full proof, refer: Full characteristicity is not upper join-closed`

### Some specific examples

One example is to take as a simple group, and as proper subgroups generating it. For instance, is the alternating group on five letters, and can be taken, for instance, as two of the embedded alternating groups on four letters.

### Another generic example

Let be a non-Abelian group generated by Abelian subgroups . Consider:

- .

Then, is the commutator subgroup both of and of , so is characteristic in both. On the other hand, is not characteristic in because the coordinate exchange automorphism (that swaps the two copies of ) does not preserve .

This generic example also shows that the property of being fully characteristic is not preserved under upper joins. `For full proof, refer: Full characteristicity is not upper join-closed`

An example is a non-Abelian group of order , and and two Abelian subgroups that generate it.