I read the following claim:

Every permutation can be expressed as the product of one and only one of the following:

an odd number of transpositions ⟺ odd permutation an even number of transpositions ⟺ even permutation

But I don't think this is correct at all, a transposition contains exactly 2 numbers so what about the following permutation: (1)(2), how can you write in the the mentioned way?

  • 6
    $\begingroup$ In this context, the identity permutation can be considered an even permutation, as it can be expressed as the product of zero transpositions, which is an even number. In general, the identity permutation is considered even, and it belongs to the alternating group as well. $\endgroup$
    – rumathe
    Mar 17 at 12:20
  • 6
    $\begingroup$ In addition to being the product of zero transpositions, the identity can also be written as $(1)(2) = (12)(12)$, which is the product of two transpositions. This is of course still even. $\endgroup$ Mar 17 at 13:06


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