Finite symmetric groups



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Finite symmetric groups

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

. Lemma   For any two finite and nonempty sets and , the natural actions of on and on are isomorphic if and only if .

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

hoping that it will be always clear from the context if this set is meant or its cardinality . It is an old tradition to prefer the set and its symmetric group which we should denote by in order to be consistent. Hence let us fix the notation for the elements of , the corresponding notation for the elements of is then obvious.

A permutation is written down in full detail by putting the images in a row under the points , say

This will be abbreviated by

Hence, for example, consists of the following elements:

In our programs permutations will be written in the form . There is a program to compute all elements of the symmetric group .

Exercises

E .   Prove lemma gif.



next up previous contents
Next: Cycle decomposition Up: Actions Previous: Paradigmatic Examples 2



Herr Fripertinger
Sun Feb 05 18:28:26 MET 1995