I am reading something about abstract algebra and got bit puzzled here.
Suppose we know every permutation is a product of transpositions (cycles with length of two). If the definition of an even permutation is that it's a product of an even number of transpositions, then isn't that true that every permutation actually correspond to a cycle that has an odd length?
For example, $(2\,3)$ is an odd permutation, while $(1\,2\,3) = (1\,2)(2\,3)$ is even permutation but it is also a cycle notation with odd length
Can any one please provide some details about this? Thank you.
Best Answer
What about the permutation $(1\,2)(3\,4)$? Not all permutations can be written as a single cycle.
However, as you say, all permutations can be written as a product of transpositions, which lends itself to the well-defined classification of a permutations being either even or odd.