Retract is transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., retract) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Suppose are groups such that is a retract of and is a retract of , then is a retract of .
Further information: Retract
A subgroup of a group is termed a retract of if there exists a homomorphism such that for all .
Given: A group , a retract of , a retract of .
To prove: is a retract of .
Proof: Let be a retraction, i.e., is a homomorphism such that for all . Let be a retraction, i.e., is a homomorphism such that for all .
Now consider the composite map . We want to argue that this is a retraction. Consider . Then, by construction , so . Since , , and so we get . Thus, is a retraction, and thus is a retract of .