Difference between revisions of "Complex"
From Online Dictionary of Crystallography
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==Definition== | ==Definition== | ||
| − | A '''complex''' is a subset obtained from a group by | + | 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 [[coset]]s. In fact, a coset does not contain the identity and therefore it is not a group. | A typical example of complexes is that of [[coset]]s. In fact, a coset does not contain the identity and therefore it is not a group. | ||
Revision as of 15:44, 15 May 2013
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 compositions for complexes.
- 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.
- 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.