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Semidirect product

  • Page ID
    19294
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    In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, one of which is normal.

    Let G be a group, N a normal subgroup of G (i.e., NG) and H a subgroup of G. G is a semidirect product of N and H if there exists a homomorphism GH which is the identity on H and whose kernel is N. This is equivalent to say that:

    • G = NH and NH = {1} (where "1" is identity element of G )
    • G = HN and NH = {1}
    • Every element of G can be written as a unique product of an element of N and an element of H
    • Every element of G can be written as a unique product of an element of H and an element of N

    One also says that "G splits over N".


    Semidirect product is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Online Dictionary of Crystallography.

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