More specifically, an extension of by is a triple where is a group, is an injective group homomorphism of into , and is a surjective homomorphism of onto such that the kernel of is the image of . Often, the extension is written as the diagram .
An extension is central if lies in the center of ; this is only possible if is commutative. It is called trivial if (the direct product of and ), is the canonical mapping of into , and is the projection homomorphism onto .
Let and be two extensions of by . A morphism of extensions from to is a homomorphism such that and .
A retraction of an extension is a homomorphism such that is the identity function on . Similarly, a section of an extension is a homomorphism such that .
Equivalence of Extensions
Theorem 1. Let and be extensions of by . Let be a morphism of extensions. Then is an isomorphism of onto . In other words, every extension morphism is an extension isomorphism.
Proof. Suppose are elements of such that . Then so . Let be an element such that . Then It follows that is the identity of , so is the identity of and .
Let be in ; since is surjective, there exists such that . Then Thus there exists such that ; then . Thus is surjective.
Theorem 2. Let be an extension of by . Then the following are equivalent:
- is the trivial extension;
- admits a retraction;
- admits a section such that lies in the centralizer of .
Proof. If is the trivial extension, then the projection onto and the canonical injection of into show that conditions 2 and 3 are satisfied. If has a retraction , then the mapping is an extension morphism, so is isomorphic to the trivial extension. If (3) holds, then the mapping is an extension morphism , so again is isomorphic to the trivial extension of by .
Note that an extension may be nontrivial, but may still be isomorphic to .