Symmetric groups are rational

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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.