# Complex

(Difference between revisions)
 Revision as of 17:13, 9 November 2017 (view source) (Added German and Spanish translations (U. Mueller))← Older edit Latest revision as of 16:27, 18 June 2019 (view source) (Lang (Fr, It)) Line 1: Line 1: - Komplex (''Ge''). Complejo (''Sp''). + Complexe (''Fr''). Komplex (''Ge''). Complesso (''It''). Complejo (''Sp''). ==Definition== ==Definition== A '''complex''' is a subset obtained from a group by choosing part of its elements in such a way that the closure property of groups is not respected. Therefore, a complex is not a group itself. A '''complex''' is a subset obtained from a group by choosing part of its elements in such a way that the closure property of groups is not respected. Therefore, a complex is not a group itself.

## Latest revision as of 16:27, 18 June 2019

Complexe (Fr). Komplex (Ge). Complesso (It). Complejo (Sp).

## Definition

A complex is a subset obtained from a group by choosing part of its elements in such a way that the closure property of groups is not respected. Therefore, a complex is not a group itself.

A typical example of complexes is that of cosets. In fact, a coset does not contain the identity and therefore it is not a group.

A subgroup is a particular case of complex that obeys the closure property and is a group itself.

## Laws of composition for complexes

There exist two laws of composition for complexes.

1. Addition. The sum of two complexes K and L consists of all the elements of K and L combined. The addition of complexes is therefore a union from a set-theoretic viewpoint. It is commutative and associative.
2. Multiplication. The product of two complexes K and L is the complex obtained by formal expansion: {KiLj}. It is, in general, non-commutative, but associative and distributive.

It is, in general, not permissible to apply the cancelling rule to complexes. This means that from the equation KL = KM does not follow that L = M, unless K is a single element.