# Polynormal subgroup

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]
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

## Definition

### Definition with symbols

A subgroup $H$ of a group $G$ is termed polynormal if given any $g \in G$, $H$ is a contranormal subgroup in the subgroup $H^{\langle g \rangle}$, i.e., the closure of $H$ under the action by conjugation of the cyclic subgroup generated by $g$.

## Metaproperties

### Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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If $H$ is polynormal in $G$, $H$ is also polynormal in any intermediate subgroup $K$. For full proof, refer: Polynormality satisfies intermediate subgroup condition

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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The whole group and the trivial subgroup are polynormal; in fact they are normal.

### Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
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ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal. For full proof, refer: Polynormality is strongly join-closed

## Testing

### GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup property at: IsPolynormal
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GAP-codable subgroup property