Symmetric group:S3

Verbal definitions
The symmetric group $$S_3$$ can be defined in the following equivalent ways:


 * It is the member of family::symmetric group on a set of three elements, viz., the group of all permutations of a three-element set. In particular, it is a member of family::symmetric group of prime degree and member of family::symmetric group of prime power degree.
 * It is the of order six (degree three), viz., the group of (not necessarily orientation-preserving) symmetries of the equilateral triangle.
 * It is the member of family::special linear group of degree two $$SL(2,2)$$ over the field of two elements. It turns out that, because of the nature of the prime two, it is also the member of family::projective special linear group of degree two $$PSL(2,2)$$, the member of family::general linear group of degree two $$GL(2,2)$$, and the member of family::projective general linear group of degree two $$PGL(2,2)$$.
 * It is the member of family::general affine group of degree one over the field of three elements, i.e., $$GA(1,3)$$ (sometimes also written as $$AGL(1,3)$$).
 * It is the member of family::general semilinear group of degree one over the field of four elements, i.e., $$\Gamma L(1,4)$$.
 * It is the member of family::von Dyck group with parameters $$(2,2,3)$$, and in particular, is a member of family::Coxeter group. In particular, it has the presentation (where $$e$$ denotes the identity element):

$$\langle a,b,c \mid a^2 = b^2 = c^3 = abc = e \rangle$$.

In the Coxeter language, this is written as:

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

Multiplication table
We portray elements as permutations on the set $$\{ 1,2,3 \}$$ using the cycle decomposition. The row element is multiplied on the left and the column element on the right, with the assumption of functions written on the left. This means that the column element is applied first and the row element is applied next.

If we used the opposite convention (i.e., functions written on the right), the row element is to be multiplied on the right and the column element on the left.

Here is the multiplication table where we use the one-line notation for permutations, where, as in the previous multiplication table, the column permutation is applied first and then the row permutation. Thus, with the left action convention, the row element is multiplied on the left and the column element on the right:

Families
The symmetric group on three elements is part of some important families:

Conjugacy class structure
As for any symmetric group, cycle type determines conjugacy class. The cycle types, in turn, are parametrized by the unordered integer partitions of $$3$$. The conjugacy classes are described below.

This group is one of three finite groups with the property that any two elements of the same order are conjugate. The other two are the cyclic group of order two and the trivial group.

Automorphism class structure
The classification of elements upto automorphism is the same as that upto conjugation; this is because the symmetric group on three elements is a complete group: a centerless group where every automorphism is inner.

Smallest of its kind

 * This is the unique non-abelian group of smallest order. All groups of order up to $$5$$, and all other groups of order $$6$$, are abelian.
 * This is the unique non-nilpotent group of smallest order. All groups of order up to $$5$$, and all other groups of order $$6$$, are nilpotent.
 * This is the unique smallest nontrivial complete group.