Semi-direct product
The (external) semi-direct product, in group theory, is a generalization of the direct product.
Two Equivalent Definitions
Let be a group,
a normal subgroup of
, and
a subgroup of
. If
and
, then
is called the (left) (external) semi-direct product of
and
.
Since is normal, the restriction of each inner automorphism of
to
is an automorphism of
. In particular, there exists a function
which associates each element of
with an automorphism on
(namely, the restriction to
of the inner automorphism on
). Then
is called the (external) semi-direct product of
by
relative to
and is denoted
. Each element of
is identified with its corresponding element of
, and the group law on
is defined as
for
Conversely, let and
be groups, and let
be a homomorphism from
into the group of automorphisms of
. The set
under the operation
is a group; it is
. Indeed,
so the law of composition is associative; the identity is
; and the inverse of
is
.
Semi-direct products and extensions
Evidently, if is a semidirect product of
by
, then it is a group extension of
by
with a section (the projection onto
). The converse is also true. Indeed, if
be an extension of
by
with a section
, then
, and
is a normal subgroup of
.