# Join of characteristic and characteristic-potentially characteristic implies characteristic-potentially characteristic

This article describes a computation relating the result of the Join operator (?) on two known subgroup properties (i.e., Characteristic subgroup (?) and Characteristic-potentially characteristic subgroup (?)), to another known subgroup property (i.e., Characteristic-potentially characteristic subgroup (?))

View a complete list of join computations

## Contents

## Statement

Suppose are subgroups of a group such that is a characteristic subgroup of and is a characteristic-potentially characteristic subgroup of . Then, the join of subgroups (which in this case is equal to the product of subgroups ) is also a characteristic-potentially characteristic subgroup.

## Definitions used

### Characteristic-potentially characteristic subgroup

`Further information: Characteristic-potentially characteristic subgroup`

## Related facts

### Similar facts

About joins:

About intersections:

- Intersection of characteristic and characteristic-potentially characteristic implies characteristic-potentially characteristic
- Intersection of characteristic and normal-potentially relatively characteristic implies normal-potentially relatively characteristic

About composition (subgroups of subgroups):

- Characteristic of potentially characteristic implies potentially characteristic
- Characteristic of characteristic-potentially characteristic implies characteristic-potentially characteristic
- Characteristic of normal-potentially characteristic implies normal-potentially characteristic
- Characteristic of normal-potentially relatively characteristic implies normal-potentially relatively characteristic

## Facts used

## Proof

**Given**: A group , subgroups of such that is characteristic in and is characteristic-potentially characteristic in .

**To prove**: The join is also characteristic-potentially characteristic in .

**Proof**: By the definition of characteristic-potentially characteristic, there is a group containing such that both and are characteristic in .

- is characteristic in : is characteristic in and is characteristic in , so fact (1) yields that is characteristic in .
- is characteristic in : and are both characteristic in , so by fact (2), so is .

Thus, is a group containing such that both and are characteristic in . Thus, is characteristic-potentially characteristic in .