Absolute Retracts in Group Theory

The subgroup R of the group G has been termed a retract of the group G whenever there exists an idempotent endomorphism of G which maps G upon R. This definition is in strict analogy to the topological concept of retract. Thus one may be tempted to define absolute retracts in like similarity to topological usage. However, we shall prove in the course of the present note that the identity is the only group which is a retract of every containing group. Consequently only modifications of the topological concept will be useful, and we shall show that each of the following classes of groups may in a certain sense lay claim to the title of absolute retract: the complete groups, the abelian groups the orders of whose elements are finite and square free, and the free groups. The following definition of the concept of retract is equivalent to the one given above, but it will be a little bit easier to handle: The group R is a retract of the group G if R is a subgroup of G, and if there exists an endomorphism e of G with the following properties: