Retract

Introduction of the concept
The concept of retract is fairly old, and came about in the beginning of the study of group theory. Retracts were first encountered as the right part in short exact sequences that split.

Introduction of the term
The term retract is not very standard, and the concept is often referred to without the use of this formal term. The term retract actually comes from the set-theoretic/topological equivalent notion.

Metaproperties
The property of being a retract is transitive. In other words, a retract of a retract is a retract. In symbols, if $$H$$ is a retract of $$G$$, and $$G$$ is a retract of $$K$$, then $$H$$ is a retract of $$K$$.

The trivial subgroup is clearly a retract, the retraction being the trivial map. The improper subgroup, viz. the whole group, is also clearly a retract, the retraction map being the identity map. Thus, the property of being a retract is trim.

Is the intersection of two retracts a retract? The answer is in general no, because it may even happen that an intersection of direct factors is not a retract. However, we do have some partial results:


 * The intersection of two retracts in a free group is a retract.