Semidirect product

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: 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 (a.k.a. splitting extension). Given a group G with identity element e, a subgroup H, and a normal subgroup N ◁ G; then the following statements are equivalent: 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 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. Let G be a semidirect product of the normal subgroup N and the subgroup H. Let Aut(N) denote the group of all automorphisms of N. The map φ: H → Aut(N) defined by φ(h) = φh, where φh is conjugation by h, thus φ(h)(n) = φh(n) = hnh−1 for all h in H and n in N, is a group homomorphism. (Note that hnh−1 ∈ N since N is normal in G.) Together N, H, and φ determine G up to isomorphism, as we show now. Given any (even unrelated) two groups N and H and a group homomorphism φ: H → Aut(N), we can construct a new group N ⋊φ H, called the (outer) semidirect product of N and H with respect to φ, defined as follows.