Splittable group
A group generated by proper subgroups
and
with
normal in
and
(so that the quotient group
is isomorphic to
, cf. Normal subgroup).
is called a split extension of the group
by the group
, or a semi-direct product of
and
. If the subgroups
and
commute elementwise, i.e.
for all
,
, their semi-direct product coincides with the direct product
. A semi-direct product
of a group
and a group
is given by a homomorphism
of
into the group
of automorphisms of
. In this case, the formula
![]() |
for all ,
, defines the multiplication in
. In the case when
and
is the identity mapping,
is called the holomorph of
(cf. Holomorph of a group).
References
[1] | D. Gorenstein, "Finite groups" , Chelsea, reprint (1980) |
Comments
Conversely, if is a semi-direct product, then conjugation with
in
defines a homomorphism
from which
can be reconstructed, i.e.
![]() |
As a set the semi-direct product of and
is
. The subsets
,
are subgroups that identify with
and
.
Splittable group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Splittable_group&oldid=35137