Generating sets for symmetric group:S3

We take $$S_3$$ as the symmetric group acting on the set $$\{ 1,2,3 \}$$.

List of generating sets
The minimum size of generating set is 2 and the maximum size of irredundant generating set is also 2. Some important generating sets are listed below.

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 assigna 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.

Bruhat ordering
The symmetric group of degree three can be viewed as a Coxeter group, with generators $$s_1 = (1,2)$$ and $$s_2 = (2,3)$$. The presentation is:

$$\langle s_1, s_2 \mid s_1^2 = e, s_2^2 = e, (s_1s_2)^3 = e \rangle$$.

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 $$\{ 1,2,3 \}$$ 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:

The element of length $$3$$, is, in matrix terms, the antidiagonal matrix:

$$\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\\end{pmatrix}$$