Complex product

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The complex product , usually simply called product , is a term from a mathematical sub-area , group theory .

Is a group and are and subsets of , then the complex product of with is defined as


There are also the abbreviations

common, where is an element of the group.

Since the above definition only requires the existence of a two-digit link , the complex product can also be viewed in more general structures, for example in semigroups .


  • The complex product of two subgroups and is a union of left subclasses of and a union of right subclasses of :
  • If and are finite subgroups, then the equation holds for the thickness of the complex product
  • The complex product of a normal subgroup with a subgroup gives a subgroup. More precisely, it holds for all subgroups and that is a subgroup if and only if applies. If or is a normal divisor, this is fulfilled. In particular, the complex product of subgroups in Abelian groups is a subgroup.
  • The complex product of secondary classes and a normal divisor is . With this product, the secondary classes of normal factors form a group, the factor group of after .
  • Is the normal divisor and subgroup of that have the properties and , then the inner semidirect product of with . For the existence of such a subgroup for a given normal subgroup, reference is made to the Schur-Zassenhaus theorem .
  • The power set of a group, together with the complex product no group, yet still Monoid , in particular the complex product is associative , so .