# Varying characteristicity

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This is a survey article related to:characteristicity
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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

## Strengthening for intermediate subgroup condition, related ideas

### 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 $p$ and outputs the property of being a subgroup $H$ of $G$ such that for any subgroup $K$ of $G$, $H \cap K$ satisfies $p$ in $K$.

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 $p$ and outputs the property of being a subgroup $H$ in a group $G$ such that for any surjective homomorphism from $G$, the image of $H$ satisfies $p$ in the image of $G$.

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