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 \(2\) which fixes all but two elements.
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 \(2\).
- Symmetric group
The symmetric groups form the fundamental link between group theory and the notion of symmetry.