Symmetric group on finite set
A symmetric group on finite set or symmetric group of finite degree is a symmetric group on a finite set.
See symmetric group for more general information about symmetric groups.
Small finite values
Since alternating groups are simple for degree at least five, all symmetric groups of degree at least five are not solvable. Also, all symmetric groups of degree greater than two are centerless, and among them, the one of degree six is the only one that is not complete.
|Cardinality of set||Common name for symmetric group of that degree||Order with prime factorization||Comments|
|2||Cyclic group:Z2||2||group of prime order. In particular, abelian|
|3||Symmetric group:S3||supersolvable but not nilpotent. Also, complete|
|4||Symmetric group:S4||solvable but not supersolvable or nilpotent. Also, complete|
|5||Symmetric group:S5||not solvable. Has simple non-abelian subgroup of index two. Also, complete|
|6||Symmetric group:S6||not solvable, and not complete.|
|7||Symmetric group:S7||complete, not solvable, has simple non-abelian subgroup of index two.|
|8||Symmetric group:S8||complete, not solvable, has simple non-abelian subgroup of index two.|
Below we discuss properties satisfied for the symmetric group of degree .
|abelian group||No for|
|nilpotent group||No for|
|solvable group||No for|
|complete group||Yes for||symmetric groups are complete|
|centerless group||Yes for||symmetric groups are centerless|
|ambivalent group||Yes||symmetric groups are ambivalent|
|strongly rational group||Yes||symmetric groups are strongly rational|