Group extension
Let and
be groups. An extension of
by
is a solution to the problem of finding a group
that contains a normal subgroup
isomorphic to
such that the quotient group
is isomorphic to
.
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
.