Symmetric group S_3 - Knowlpedia

The symmetric group $S_3$ is the group of all bijections of $\{1,\ldots,3\}$ , with composition as multiplication.

Order and normal subgroup

There are 6 permutations: choose the images of the 3 letters successively. The even permutations form the proper nontrivial normal subgroup $A_3$ , the kernel of the sign map. Therefore $S_3$ is not simple.

Triangle symmetries

The generators $(123)$ and $(12)$ satisfy the dihedral relations. The action on three vertices identifies this group with $D_6$ .

References

J. S. Milne, Group Theory, v4.01, §4, symmetric and alternating groups; Theorem 4.33 and Remark 4.34, p. 69.