Transposition (group theory)

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Routine for first years, needed for some manifolds work.

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Definition

Let [ilmath]S_k[/ilmath] denote the symmetric group on [ilmath]k\in\mathbb{N} [/ilmath] symbols. Then:

• [ilmath](i j)[/ilmath] denotes (in ordinary cycle notation the permutation:
  • [ilmath](i j): i\mapsto j[/ilmath], [ilmath](i j):j\mapsto i[/ilmath] and for all other [ilmath]m\in\{1,\ldots,k\} [/ilmath] (so [ilmath]m\ne i[/ilmath] and [ilmath]m\ne j[/ilmath]) we have: [ilmath](i j):m\mapsto m[/ilmath]

Such an [ilmath](i j)[/ilmath] is called a transposition.

See also

• Every element of the symmetric group can be written as an even or odd number of transpositions, but not both