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Concept
Semidirect product
Summary
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
an inner semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup.
an outer semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation.
As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products.
For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension).
Given a group G with identity element e, a subgroup H, and a normal subgroup N ◁ G, the following statements are equivalent:
G is the product of subgroups, G = NH, and these subgroups have trivial intersection: N ∩ H = .
For every g ∈ G, there are unique n ∈ N and h ∈ H such that g = nh.
For every g ∈ G, there are unique n ∈ N and h ∈ H such that g = hn.
The composition π ∘ i of the natural embedding i: H → G with the natural projection π: G → G/N is an isomorphism between H and the quotient group G/N.
There exists a homomorphism G → H that is the identity on H and whose kernel is N. In other words, there is a split exact sequence
of groups (which is also known as group extension of by ).
If any of these statements holds (and hence all of them hold, by their equivalence), we say G is the semidirect product of N and H, written
or
or that G splits over N; one also says that G is a semidirect product of H acting on N, or even a semidirect product of H and N. To avoid ambiguity, it is advisable to specify which is the normal subgroup.
If , then there is a group homomorphism given by , and for , we have .
Let us first consider the inner semidirect product.
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