Symmetric group: Difference between revisions
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===Weaker properties=== | ===Weaker properties=== | ||
* [[Centerless group]]: All except the symmetric group on two elements are centerless | * [[Centerless group]]: All except the symmetric group on two elements are centerless. {{further|[[Symmetric groups are centerless]]}} | ||
* [[Complete group]]: All except the symmetric group on two elements and the symmetric group on six elements | * [[Complete group]]: All except the symmetric group on two elements and the symmetric group on six elements. {{further|[[Symmetric groups are complete]]}} | ||
==Related notions== | |||
===Alternating group=== | |||
===Finitary symmetric and alternating groups=== | |||
===Subgroups of the symmetric group=== | |||
==IAPS structure== | ==IAPS structure== | ||
{{further|[[Permutation IAPS]]}} | {{further|[[Permutation IAPS]]}} | ||
Revision as of 10:37, 16 June 2008
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
To get more information on particular symmetric groups, check out Category:Symmetric groups
Definition
Symbol-free definition
The symmetric group on a set is the group of all permutations of that set. A group is said to be a symmetric group if it is isomorphic to the symmetric group on some set.
Definition with symbols
The symmetric group over a set (denoted as ) is defined as the group of all permutations on , with the multiplication being function composition.
A group is termed a symmetric group if for some set .
Relation with other properties
Weaker properties
- Centerless group: All except the symmetric group on two elements are centerless. Further information: Symmetric groups are centerless
- Complete group: All except the symmetric group on two elements and the symmetric group on six elements. Further information: Symmetric groups are complete
Related notions
Alternating group
Finitary symmetric and alternating groups
Subgroups of the symmetric group
IAPS structure
Further information: Permutation IAPS