Definition 9

Tag 05BQ, permanent. Cite it as [Bourbaki, Tag 05BQ].

Algebra, chapter I, § 5. Groups operating on a set

A cycle of order 2 is called a transposition.

Let $x$ and $y$ be two distinct elements of E. Let $\tau_{x,y}$ denote the unique transposition with support $\{x, y\}$ .

For every permutation $\sigma$ of E the permutation $\sigma \cdot \tau_{x,y} \cdot \sigma^{-1}$ is a transposition whose support is $\{\sigma(x), \sigma(y)\}$ . Hence:

(4)

\sigma \cdot \tau_{x,y} \cdot \sigma^{-1} = \tau_{\sigma(x), \sigma(y)}.

Transpositions thus form a conjugacy class in the group $\mathfrak{S}_E$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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