Table of Contents - 10.2.10.4. Transpositions in the symmetric group
Transpositions are special cases of "cycles" as defined in [Rotman] p. 28: "Let
i<sub>1</sub> , i<sub>2</sub> , ... , i<sub>r</sub> be distinct integers
between 1 and n. If α in S<sub>n</sub> fixes the other integers and
α(i<sub>1</sub>) = i<sub>2</sub>, α(i<sub>2</sub>) = i<sub>3</sub>,
..., α(i<sub>r-1</sub> ) = i<sub>r</sub>, α(i<sub>r</sub>) =
i<sub>1</sub>, then α is an <b>r-cycle</b>. We also say that α is a
cycle of <b>length</b> r." and in [Rotman] p. 31: "A 2-cycle is also called
<b>transposition</b>.".
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We (currently) do not have/need a definition for cycles, so transpositions are
explicitly defined in df-pmtr.