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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.

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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.

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As Mike Pierce notes, not all permutations can be written as a cycle at all.

However, it's true that under this definition, odd cycles are even and even cycles are odd. To avoid confusing oneself, it would be good to distinguish between an odd cycle and an odd permutation. Thus for example, $(123) = (13)(12)$ is on the one hand an odd cycle and on the other hand an even permutation.

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