Two Equivalent Definitions
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 .