Symmetric groups are rational
From Groupprops
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
- Finitary symmetric group on infinite set is rational
- Finitary alternating group on infinite set is rational
- Symmetric groups are ambivalent
- Symmetric groups are strongly ambivalent
Facts used
Proof
Proof outline
- Take any permutation
. Express it using its cycle decomposition.
- Show that any other permutation
generating the same cyclic group has the same cycle type as
.
- Use the fact that any two permutations with the same cycle type, are conjugate inside the symmetric group.