Complex

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: {K_iL_j}. 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.