Finite symmetric groups

Finite symmetric groups

In the first section we mentioned the symmetric group S_X on the set X. In order to prepare further examples and detailed descriptions of actions we need to consider this group in some detail, in particular for finite X. A first remark shows that it is only the order of X which really matters:

Lemma: For any two finite and nonempty sets X and Y, the natural actions of S_X on X and S_Y on Y are isomorphic if and only if | X | = | Y | .

This is very easy to check and therefore left as an exercise. We call | X | the degree of S_X, of any subgroup P £S_X and of any pÎS_X. In order to examine permutations of degree n it therefore suffices to consider a particular set of order n and its symmetric group. For technical reasons we introduce two such sets of order n:

n:= {0, ...,n-1 } and n:= {1, ...,n },

hoping that it will be always clear from the context if this set n is meant or its cardinality n. It is an old tradition to prefer the set n and its symmetric group which we should denote by S_n in order to be consistent. Hence let us fix the notation for the elements of S_n , the corresponding notation for the elements of S_n is then obvious. A permutation pÎS_n is written down in full detail by putting the images pi in a row under the points i În, say This will be abbreviated by Hence, for example, S ₃ consists of the following elements: In our programs permutations will be written in the form [ p1, p2, ..., pn]. There is a program to compute all elements of the symmetric group S_n .

• Cycle decomposition
• Conjugation
• Conjugacy Classes
• Generators of the symmetric group
• Subgroups of Cyclic Groups
• Colourings of the n Gon
• The Sign
• Splitting Orbits
• Rothe diagram and inversions
• The Lehmer code and reduced decompositions
• Sn as a Coxeter Group
• The Exchange Lemma
• Exercises

Finite symmetric groups