# Complex product

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 .

## properties

- 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 .

## literature

- Thomas W. Hungerford:
*Algebra*. 5. print. Springer-Verlag, 1989, ISBN 0-387-90518-9