Semidirect Product
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](/web/20201020192217im_/https://mathworld.wolfram.com/images/equations/SemidirectProduct/Inline10.gif)
(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).
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