Transposition (group theory)

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Routine for first years, needed for some manifolds work.

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Definition

Let $S_k$ denote the symmetric group on $k\in\mathbb{N}$ symbols. Then:

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

Such an $(i j)$ 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