# Finite direct power-closed characteristic is quotient-transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., finite direct power-closed characteristic subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about finite direct power-closed characteristic subgroup |Get facts that use property satisfaction of finite direct power-closed characteristic subgroup | Get facts that use property satisfaction of finite direct power-closed characteristic subgroup|Get more facts about quotient-transitive subgroup property

## Statement

### Statement with symbols

Suppose is a group and and are subgroups with . Suppose is a finite direct power-closed characteristic subgroup of . Since characteristic implies normal, we can talk of the quotient group . Suppose, further, that is a finite direct power-closed characteristic subgroup of . Then, is also a finite direct power-closed characteristic subgroup of .

## Facts used

- Characteristicity is quotient-transitive: If with characteristic in and characteristic in , then is characteristic in .
- (no link): The fact that and , and the natural embedding from to coincides via these isomorphisms with the natural embedding from to . where stands for the direct power.

## Proof

**Given**: A group , a finite direct power-closed characteristic subgroup of . A subgroup of containing such that is a finite direct power-closed characteristic subgroup of . is a natural number.

**To prove**: is a characteristic subgroup of under the natural embedding.

**Proof**:

Step no. | Assertion | Definitions used | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|---|

1 | is characteristic in | finite direct power-closed characteristic subgroup | -- | is finite direct power-closed characteristic in | -- | Piece together definition and given data. |

2 | is characteristic in | -- | fact (2) | -- | step (1) | Using the natural identification indicated by fact (2), step (2) becomes a reformulation of step (1). |

3 | is characteristic in | finite direct power-closed characteristic subgroup | -- | is finite direct power-closed characteristic in | -- | Piece together definition and given data. |

4 | is characteristic in | -- | fact (1) | -- | steps (2) and (3) | Piece together. [SHOW MORE] |