Generating sets for symmetric group:S3
This article gives specific information, namely, generating sets, about a particular group, namely: symmetric group:S3.
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We take as the symmetric group acting on the set .
List of generating sets
|Generating set||Size of generating set||Diameter||Statistics for number of group elements for each minimum word length, starting from zero and going up to the diameter (must add up to 6)||Other similar generating sets under conjugation||More information|
|3||2||1,3,2||none||#Generating set of all transpositions|
|2||2||1,3,2||#Dihedral rotation and reflection generating set|
Generating set of all transpositions
Note that the left and right Cayley graphs are identical because the generating set is a conjugacy class of involutions. Also, we can unambiguously assign a direction (away from the identity) to each edge because there are no cycles of odd length, which in turn follows from the fact that all the generators are odd permutations.
The symmetric group of degree three can be viewed as a Coxeter group, with generators and . The presentation is:
We can thus consider a Bruhat ordering on the elements of the symmetric group of degree three. Note that the Bruhat ordering depends on the specific choice of transpositions we use to generate the group, which in turn depends on an implicit order of the elements that the group acts on (up to reversal). Thus, the Bruhat ordering is not invariant under conjugation.
The Bruhat ordering on the symmetric group of degree three has the special feature (no longer true for higher degree) that any two elements with distinct Bruhat lengths are comparable in the order. In the Bruhat ordering, there are four levels based on Bruhat length:
|Length||Number of elements of that length||Elements of that length||Conjugacy class information for these elements|
|0||1||-- the identity element||a single conjugacy class|
|1||2||and||all the elements are conjugate but do not form a complete conjugacy class|
|2||2||and||the elements form a single conjugacy class|
|3||1||a single element, part of a conjugacy class whose other elements have length 1|
The element of length , is, in matrix terms, the antidiagonal matrix: