Example:Permutation (group theory) of S5

{{Stub page|grade=A*|msg=Add context, this is an example that is used on the Symmetric group page itself. That provides the context]]

Context

TODO: Do context

Terms exemplified

• Cycle notation
• Transposition
• Symmetric group

Example

Let us consider [ilmath]S_5[/ilmath] as an example.

• Let [ilmath]\sigma\in S_5[/ilmath] be the permutation given as follows:
  • [ilmath]\sigma:1\mapsto 3[/ilmath], [ilmath]\sigma:2\mapsto 2[/ilmath], [ilmath]\sigma:3\mapsto 5[/ilmath], [ilmath]\sigma:4\mapsto 1[/ilmath], [ilmath]\sigma:5\mapsto 4[/ilmath]

  This can be written more neatly as:

  • [math]\left(\begin{array}{ccccc}1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 5 & 1 & 4\end{array}\right)[/math], the thing in the top row is sent to the thing below it.
    • This can be written as the product of disjoint cycles too:
      • [math](1\ 3\ 5\ 4)[/math] or [ilmath](1\ 3\ 5\ 4)(2)[/ilmath] if you do not take the "implicit identity" part. That is any element not in a cycle stays the same
    • Or as transpositions
      • [math](1\ 4)(1\ 5)(1\ 3)[/math] - recall we read right-to-left, so this is read:
        • [ilmath]1\mapsto 3\mapsto 3\mapsto 3[/ilmath]
        • [ilmath]3\mapsto 1\mapsto 5\mapsto 5[/ilmath]
        • [ilmath]5\mapsto 5\mapsto 1\mapsto 4[/ilmath]
        • [ilmath]4\mapsto 4\mapsto 4\mapsto 1[/ilmath] - the cycle [ilmath](1\ 3\ 5\ 4)[/ilmath]
        • And of course [ilmath]2\mapsto 2\mapsto 2\mapsto 2[/ilmath]

Discussion

Grade: C

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