 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. == some equivalent definitions of inner semidirect products == let g be a group with identity element e, a subgroup h and a normal subgroup n (i.e., ). with this premise, the following statements are equivalent: * g = nh and n â© h = {e}. * every element of g can be written in a unique way as a product nh, with  and . * every element of g can be written in a unique way as a product hn, with  and . * the natural embedding , composed with the natural projection , yields an isomorphism between h and the quotient group . * there exists a homomorphism  that is the identity on h and whose kernel is n. if one (and therefore all) of these statements hold, we say that g is a semidirect product of n and h, written :g = n \rtimes h, 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 of the two subgroups is normal. == elementary facts and caveats == if g is the semidirect product of the normal subgroup n and the subgroup h, and both n and h are finite, then the order of g equals the product of the orders of n and h. note that, as opposed to the case with the direct product, a semidirect product of two groups is not, in general, unique; if g and gâ² are two groups which both contain isomorphic copies of n as a normal subgroup and h as a subgroup, and both are a semidirect product of n and h, then it does not follow that g and gâ² are isomorphic. this remark leads to an extension problem, of describing the possibilities. == semidirect products and group homomorphisms == let g be a semidirect product of the normal subgroup n and the
Group_theory