semidirect product

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semidirect product

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Refers to the fact that the criteria are less strict than for the direct product; compare semidirect.

semidirect product (plural semidirect products)

1. (group theory) A generalisation of direct product such that, in one of two equivalent definitions, only one of the subgroups involved is required to be a normal subgroup.
   • 1998, Hernán Cendra, Darryl D. Holm, Jerrold E. Marsden, Tudor S. Ratiu, Lagrangian Reduction, the Euler-Poincaré Equations, and Semidirect Products, A. G. Khovanskiĭ, A. Varchenko, V. Vassiliev (editors), Geometry of Differential Equations, American Mathematical Society, Translations, Series 2, Volume 186, Advances in the Mathematical Sciences 39, page 8,

     The preceding result is a special case of a general theorem on reduction by stages for semidirect products acting on a symplectic manifold .

   • 2008, I. Martin Isaacs, Finite Group Theory, American Mathematical Society, page 69:

     Also, by Lemma 3.1, every group G with a normal subgroup N and complement H is isomorphic to a semidirect product of N by H, and so once we prove Theorem 3.2, it will be fair to say that we have constructed all possible split extensions (up to isomorphism).

   • 2012, Fernando Q. Gouvêa, A Guide to Groups, Rings, and Fields, page 61:

     Theorem 4.8.5 Let ${\displaystyle G}$ be a group, ${\displaystyle H,K<G,H}$◅ ${\displaystyle G}$. Then ${\displaystyle G}$ is the internal semidirect product of ${\displaystyle H}$ and ${\displaystyle K}$ if and only if ${\displaystyle H\cap K=1}$ and ${\displaystyle G=HK}$.
     Many groups can be understood as semidirect products.

Two equivalent mathematical definitions exist, which, for didactic purposes, are sometimes distinguished as inner semidirect product and outer semidirect product.

• inner semidirect product
• outer semidirect product