Symmetric group:S5

Definition
The symmetric group $$S_5$$ is defined in the following equivalent ways:


 * 1) It is the group of all permutations on a set of five elements, i.e., it is the member of family::symmetric group  of degree five. In particular, it is a member of family::symmetric group of prime degree and member of family::symmetric group of prime power degree. With this interpretation, it is denoted $$S_5$$ or $$\operatorname{Sym}(5)$$.
 * 2) It is the member of family::projective general linear group of degree two over the field of five elements, i.e., $$PGL(2,5)$$.
 * 3) It is the member of family::projective semilinear group of degree two over the field of four elements, i.e., $$P\Gamma L(2,4)$$.

Equivalence of definitions

 * PGammaL(2,4) is isomorphic to S5
 * PGL(2,5) is isomorphic to S5

Upto conjugacy
For convenience, we take the underlying set to be $$\{ 1,2,3,4,5 \}$$.

There are seven conjugacy classes, corresponding to the unordered integer partitions of $$5$$ (for more information, refer cycle type determines conjugacy class). We use the notation of the cycle decomposition for permutations:

Upto automorphism
$$S_5$$ is a complete group: in particular, every automorphism of the group is inner. Thus, the equivalence classes under automorphisms are the same as the conjugacy classes. In fact, $$S_n$$ is complete for $$n \ne 2, 6$$. See symmetric groups on finite sets are complete.

Automorphisms
Since $$S_5$$ is a complete group, it is isomorphic to its automorphism group, where each element of $$S_5$$ acts on $$S_5$$ by conjugation.