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).
|||D. Gorenstein, "Finite groups" , Chelsea, reprint (1980)|
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://www.encyclopediaofmath.org/index.php?title=Splittable_group&oldid=35137