# Group in which every retract is regular

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## Definition

A group in which every retract is regular is a group in which every retract is a regular retract.

## Definitions used

Term Definition
Retract A subgroup $H$ of a group $G$ is a retract if there exists a normal subgroup $N$ of $G$ such that $NH = G$ and $N \cap H$ is trivial.
Regular retract A subgroup $H$ of a group $G$ is a regular retract if there exists a subgroup $K$ of $G$ such that $\langle H, K \rangle = G$, $H$ intersects the normal closure of $K$ trivially and $K$ intersects the normal closure of $H$ trivially.

## Formalisms

### In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (retract) satisfies the second property (regular retract), and vice versa.
View other group properties obtained in this way

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Group in which every retract is a direct factor |FULL LIST, MORE INFO
Group in which every retract is a free factor every retract is a free factor |FULL LIST, MORE INFO