Semi-direct product
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The (external) semi-direct product, in group theory, is a generalization of the direct product.
Two Equivalent Definitions
Let 
be a group, 
a normal subgroup of , and 
a subgroup of . If 
and 
, then is called the (left) (external) semi-direct product of and .
Since is normal, the restriction of each inner automorphism of to is an automorphism of . In particular, there exists a function 
which associates each element of with an automorphism on (namely, the restriction to of the inner automorphism on ). Then is called the (external) semi-direct product of by relative to and is denoted 
. Each element of is identified with its corresponding element of 
, and the group law on is defined as
for
Conversely, let and be groups, and let be a homomorphism from into the group of automorphisms of . The set under the operation
is a group; it is . Indeed,
so the law of composition is associative; the identity is 
; and the inverse of 
is 
.
Semi-direct products and extensions
Evidently, if is a semidirect product of by , then it is a group extension of by with a section (the projection onto ). The converse is also true. Indeed, if 
be an extension of by with a section 
, then 
, and 
is a normal subgroup of .
