# Transposition (as an element of a symmetric group)

In a symmetric group, a transposition is a permutation which has the effect of swapping two elements while leaving everything else unchanged. More formally, it is a permutation of order $$2$$ which fixes all but two elements.

knows-requisite(Cycle type of a permutation): A transposition is precisely an element with cycle type $$2$$.

# Example

In $$S_5$$, the permutation $$(12)$$ is a transposition: it swaps $$1$$ and $$2$$ while leaving all three of the elements $$3,4,5$$ unchanged. However, the permutation $$(124)$$ is not a transposition, because it has order $$3$$, not order $$2$$.

Parents:

• Symmetric group

The symmetric groups form the fundamental link between group theory and the notion of symmetry.