Group in which every retract is a free factor
From Groupprops
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
Contents
Definition
A group in which every retract is a free factor is a group with the property that every retract of the group is a free factor of the group, i.e., a factor in an internal free product.
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 |
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 |
