# Finite normal implies amalgam-characteristic

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., finite normal subgroup) must also satisfy the second subgroup property (i.e., amalgam-characteristic subgroup)

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finite group) must also satisfy the second group property (i.e., always amalgam-characteristic group)

View all group property implications | View all group property non-implications

Get more facts about finite group|Get more facts about always amalgam-characteristic group

## Contents

## Statement

### Verbal statement

Any finite normal subgroup (i.e., a normal subgroup that is finite as a group) is an amalgam-characteristic subgroup.

Equivalently, a finite group is an always amalgam-characteristic group: it is an amalgam-characteristic subgroup in any group in which it is a normal subgroup.

### Statement with symbols

Suppose is a finite normal subgroup of a group . Then, is a characteristic subgroup inside the amalgam .

## Related facts

### Stronger facts

### Similar facts

- Central implies amalgam-characteristic
- Periodic normal implies amalgam-characteristic
- Normal subgroup contained in hypercenter is amalgam-characteristic

### Opposite facts

- Normal not implies amalgam-characteristic
- Characteristic not implies amalgam-characteristic
- Direct factor not implies amalgam-characteristic
- Cocentral not implies amalgam-characteristic

### Applications

- Finite normal implies potentially characteristic
- Finite normal implies image-potentially characteristic

## Facts used

- Normality is upper join-closed
- Quotient of amalgamated free product by amalgamated normal subgroup equals free product of quotient groups
- Free product of nontrivial groups has no nontrivial finite normal subgroup
- Normality satisfies image condition

## Proof

**Given**: A group , a finite normal subgroup of . .

**To prove**: is a characteristic subgroup in .

**Proof**:

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

1 | is a finite normal subgroup of . | Fact (1) | is finite and normal in | Since is normal in both copies of , it is normal in . Also, is finite. | |

2 | . | Fact (2) | is normal in | Fact-direct | |

3 | has no nontrivial finite normal subgroup. | Fact (3) | Step (2) | [SHOW MORE] | |

4 | is the unique largest finite normal subgroup of . | Fact (4) | Steps (1), (3) | [SHOW MORE] | |

5 | is characteristic in . | Step (4) | [SHOW MORE] |

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format