Group in which every retract is a direct factor

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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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

A group in which every retract is a direct factor is a group satisfying the following equivalent conditions:

  1. Every retract of the group (i.e., every subgroup that has a normal complement) is a direct factor.
  2. Every retract of the group is a normal subgroup.

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 (direct factor), 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
Simple group nontrivial; no proper nontrivial normal subgroup |FULL LIST, MORE INFO
Splitting-simple group nontrivial; no proper nontrivial retract |FULL LIST, MORE INFO
Abelian group any two elements commute |FULL LIST, MORE INFO

Weaker properties

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