# Amalgam-characteristic subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

### Definition with symbols

Suppose is a group and is a subgroup of . We say that is an **amalgam-characteristic subgroup** of if is characteristic in the group given by:

.

In other words, is the amalgam of with itself over , and is treated as the subgroup of given by the amalgamated between the two factors.

## Relation with other properties

### Stronger properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal subgroup | amalgam-characteristic implies normal | normal not implies amalgam-characteristic | |FULL LIST, MORE INFO |

### Incomparable properties

- Characteristic subgroup:
`For full proof, refer: Characteristic not implies amalgam-characteristic`Note that the other non-implication is clear from, for instance, finite normal implies amalgam-characteristic, because not every finite normal subgroup is characteristic (`For full proof, refer: Normal not implies characteristic`).