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: an inner semidirect product is a particular way in which a group can be constructed from two subgroups, one of which is a normal subgroup, while an outer semidirect product is a Cartesian product as a set, but with 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 (a.k.a. split[ting] 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 these statements hold, 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) = φ_h, conjugation by h, where φ(h)(n) = φ_h(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.

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