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

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Terms exemplified

• Cycle notation
• Transposition
• Symmetric group

Example

Let us consider $S_5$ as an example.

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

  This can be written more neatly as:

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

Discussion

Grade: C

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