Symmetric groups are rational: Difference between revisions

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==Statement==
==Statement==


The [[fact about::symmetric group]] on a finite set is a [[fact about::rational group]], i.e., it satisfies the following equivalent conditions:
The [[fact about::symmetric group]] on any set (finite or infinite) is a [[fact about::rational group]].
 
# If <math>r</math> is relatively prime to the order of the group, then for any <math>g</math> in the group, <math>g</math> and <math>g^r</math> are conjugate.
# Every character of the group is rational-valued.
# Every character of the group is integer-valued.
 
==Definitions used==
 
===Symmetric group===
{{further|[[symmetric group]]}}
 
===Rational group===
{{further|[[rational group]]}}


==Related facts==
==Related facts==
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* [[Finitary symmetric group on infinite set is rational]]
* [[Finitary symmetric group on infinite set is rational]]
* [[Finitary alternating group on infinite set is rational]]
* [[Finitary alternating group on infinite set is rational]]
* [[Symmetric group on infinite set is rational]]
* [[Symmetric groups are ambivalent]]
* [[Symmetric groups are strongly ambivalent]]
 
==Facts used==
==Facts used==


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# Take any permutation <math>g</math>. Express it using its cycle decomposition.
# Take any permutation <math>g</math>. Express it using its cycle decomposition.
# Show that if <math>r</math> is relatively prime to the order of the group, then <math>g</math> and <math>g^r</math> have the same cycle type.
# Show that any other permutation <math>h</math> generating the same cyclic group has the same cycle type as <math>g</math>.
# Use the fact that any two permutations with the same cycle type, are conjugate inside the symmetric group.
# Use the fact that any two permutations with the same cycle type, are conjugate inside the symmetric group.

Latest revision as of 20:06, 3 September 2009

This article describes a basic fact about permutations, or about the symmetric group or alternating group.
View a complete list of basic facts about permutations

Statement

The Symmetric group (?) on any set (finite or infinite) is a Rational group (?).

Related facts

Facts used

  1. Cycle decomposition theorem
  2. Cycle type determines conjugacy class

Proof

Proof outline

  1. Take any permutation g. Express it using its cycle decomposition.
  2. Show that any other permutation h generating the same cyclic group has the same cycle type as g.
  3. Use the fact that any two permutations with the same cycle type, are conjugate inside the symmetric group.
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