Semidirect product

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In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product. A semidirect product is a cartesian product as a set, but with a particular multiplication operation.

Contents

Some equivalent definitions

Let G be a group with identity element e, N a normal subgroup of G (i.e., NG) and H a subgroup of G. The following statements are equivalent:

  • G = NH and NH = {e}.
  • G = HN and NH = {e}.
  • 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.
  • The natural embedding HG, composed with the natural projection GG / N, yields an isomorphism between H and the quotient group G / N.
  • There exists a homomorphism GH which 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". In order to avoid ambiguities, 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.

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