# Varying characteristicity

This is a survey article related to:characteristicity

View other survey articles about characteristicity

Characteristicity is a pivotal group property, that, alongwith normality, dates back to before the twentieth century. The notion was introduced by Frobenius for those subgroups that are fixed by *all* automorphisms of the group (and not just the inner ones).

This article surveys some of the more common among the variations of the subgroup property of characteristicity. The ideas are:

- Emulate the strengths
- Remedy the weaknesses

## Contents

### Intermediately characteristic subgroup

The intermediately operator takes as input a subgroup property and outputs the property of being a subgroup that has the original propoerty in every intermediate subgroup.

Note that a subgroup property is fixed under the intermediately operator if and only if it satisfies the intermediate subgroup condition.

The subgroup property of being characteristic does *not* satisfy the intermediate subgroup condition. Applying the intermediately operator to it gives the subgroup property of being intermediately characteristic.

### Transfer-closed characteristic subgroup

The transfer-closure operator takes as input a subgroup property and outputs the property of being a subgroup of such that for any subgroup of , satisfies in .

The transfer-closure operator applied to the subgroup property of being characteristic is the subgroup property of being transfer-closed characteristic.

### Image-closed characteristic subgroup

The image-closure operator takes as input a subgroup property and outputs the property of being a subgroup in a group such that for any surjective homomorphism from , the image of satisfies in the image of .

Applying the image-closure operator to the subgroup property of being characteristic gives the subgroup property of being image-closed characteristic.