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=48785