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
.
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