Symmetric group:S4

Definition
The symmetric group $$S_4$$ or $$\operatorname{Sym}(4)$$, also termed the symmetric group of degree four, is defined in the following equivalent ways:


 * The group of all permutations, i.e., the member of family::symmetric group on a set of size four. In particular, it is a member of family::symmetric group of prime power degree.
 * The member of family::triangle group (not the von Dyck group, but its double) $$\Delta(2,3,3)$$. In other words, it has the presentation (where $$e$$ denotes the identity element):

$$\langle s_1, s_2, s_3 \mid s_1^2 = s_2^2 = s_3^2 = e, (s_1s_2)^2 = (s_2s_3)^3 = (s_1s_3)^3 = e \rangle$$.

In particular, it is a member of family::Coxeter group.


 * The full tetrahedral group: The group of all (not necessarily orientation-preserving) symmetries of the regular tetrahedron. This is denoted as $$T_h$$.
 * The member of family::von Dyck group with parameters $$(2,3,4)$$ (sometimes written in reverse order as $$(4,3,2)$$). In other words, it has the presentation (with $$e$$ denoting the identity element):

$$\langle a,b,c \mid a^2 = b^3 = c^4 = abc = e \rangle$$.


 * The octahedral group or cube group: group of orientation-preserving symmetries of the cube (or equivalently, the octahedron). This is denoted as $$O$$.
 * The member of family::projective general linear group of degree two over the field of three elements: $$PGL(2,3)$$.
 * The member of family::general affine group of degree two over the field of two elements: $$GA(2,2)$$.
 * The member of family::projective special linear group of degree two over ring:Z4, the ring $$\mathbb{Z}/4\mathbb{Z}$$ of the integers modulo 4: $$PSL(2,\mathbb{Z}/4\mathbb{Z})$$.
 * The member of family::general semiaffine group of degree one over field:F4, i.e., the group $$\Gamma A(1,4)$$ or $$A \Gamma L(1,4)$$.

Equivalence of definitions
The following is a list of proofs of the equivalence of various definitions:


 * Full tetrahedral group is isomorphic to S4
 * von Dyck group with parameters (2,3,4) is isomorphic to S4
 * PGL(2,3) is isomorphic to S4
 * GA(2,2) is isomorphic to S4
 * PSL(2,Z4) is isomorphic to S4

Automorphisms
Since $$S_4$$ is a complete group, it is isomorphic to its automorphism group, where each element of $$S_4$$ acts on $$S_4$$ by conjugation. In fact, for $$n \ne 2,6$$, the symmetric group $$S_n$$ is a complete group.

Endomorphisms
$$S_4$$ admits four kinds of endomorphisms (that is, it admits more endomorphisms, but any endomorphism is equivalent via an automorphism to one of these four):


 * The endomorphism to the trivial group
 * The identity map
 * The retraction to a group of order two, given by the sign homomorphism.
 * The retraction to a symmetric group on three of the elements, with kernel being the Klein four-group comprising the identity element and the double transpositions. (Note that all such retractions are equivalent, and there are other equivalent endomorphisms obtained by composing such a retraction with an automorphism).

Conjugacy class structure
There are five conjugacy classes, corresponding to the cycle types, because cycle type determines conjugacy class. Further, each cycle type corresponds to a partition of $$4$$.

Automorphism class structure
Since $$S_4$$ is a complete group, all its automorphisms are inner automorphisms, and, in particular, the classification of elements up to conjugacy is the same as the classification up to automorphisms.

Supergroups
The symmetric group $$S_4$$ is contained in higher symmetric groups, most notably the symmetric group on five elements $$S_5$$.

Extensions
These include $$GL(2,3)$$ whose inner automorphism group is $$S_4$$ (specifically $$S_4$$ is the quotient of $$GL(2,3)$$ by its scalar matrices).