A "split" extension 
of groups 
and 
which contains a subgroup 
isomorphic to 
with 
and 
(Ito 1987, p. 710). Then the semidirect
product of a group 
by a group 
, denoted ![H×AdjustmentBox[│, BoxMargins -> {{-0.27, 0.13913}, {-0.5, 0.5}}]G](/images/equations/SemidirectProduct/Inline10.svg)
(or sometimes 
)
with homomorphism 
is given by
 |
where 
,
,
and 
(Suzuki 1982, p. 67; Scott 1987, p. 213). Note that the semidirect product
of two groups is not uniquely defined.
The semidirect product of a group 
by a group 
can also be defined as a group 
which is the product of its subgroups 
and 
, where 
is normal in 
and 
. If 
is also normal in 
, then the semidirect product becomes a group
direct product (Shmel'kin 1988, p. 247).
