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Transposition
A cycle of length two is called a transposition. It is a 2-cycle. For example (1 2), (2 3), (1 3) are three transpositions in S3.

Here (2, 3) is a 2-cycle such that image of 2 is 3 and the image of 3 is 2 and the remaining missing elements are fixed or invariant.

Alternative definition of transposition

A transposition is a permutation ƒ that moves exactly two elements.

Suppose that ƒ moves a, b, in which case (a) ƒ = b and (b) ƒ = a.

Thus ƒ has the form



For brevity we write

ƒ = (a, b) or ƒ = (b, a)

Rather than the fuller function table we have used heretofore. Note that a transposition ƒ is an element of order 2:

ƒ2 = I ƒ-1 = ƒ.

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