Symmetric group

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Note: the symmetric group is a permutation group on finitely many symbols, see permutation group (which uses the same notation) for the more general case.

Definition

Let $k\in\mathbb{N}$ be given. The symmetric group on $k$ symbols, denoted $S_k$, is the permutation group on $\{1,2,\ldots,k-1,k\}\subset\mathbb{N}$. The set of the group is the set of all permutations on $\{1,2,\ldots,k-1,k\}$. See proof that the symmetric group is actually a group for details.

• Identity element: $e:\{1,\ldots,k\}\rightarrow\{1,\ldots,k\}$ which acts as so: $e:i\mapsto i$ - this is the identity permutation, it does nothing.
• The group operation is ordinary function composition, for $\sigma,\tau\in S_k$ we define:
  • $\sigma\tau:\eq \sigma\circ\tau$ with: $\sigma\tau:\{1,\ldots,k\}\rightarrow\{1,\ldots,k\}$ by $\sigma\tau:i\mapsto\sigma(\tau(i))$
    • Caveat:Be careful, a lot of authors (Allenby, McCoy to name 2) go left-to-right and write $i\sigma$ for what we'd use $\sigma(i)$ or $\sigma i$ at a push for. Then $\sigma\tau$ would be $\tau\circ\sigma$ in our notation

See also

• Cycle notation
  • Every element of the symmetric group can be written as the product of disjoint cycles
  • Transposition - a $2$-cycle.
    • Every element of the symmetric group can be written as an even or odd number of transpositions, but not both