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Exercise (Practice With Even Permutations)

Show that each permutation below is an even permutation.

  1. $(1,2)(3,4)$
  2. $(1,2,3)$
  3. $(1,2)(3,4)(1,2,3)$
  4. $()$
  5. $(1,4,3,5)(2,3,1,4,7,6)$

Click to see a solution.

With each permutation, we just have two show that when we write the permutation as a product of transpositions, that the number of them is even.

  1. The permutation $(1,2)(3,4)$ is already written as a product of two transpositions, so it's an even permutation.
  2. We can write $(1,2,3)=(1,3)(1,2)$, which is two transpositions.
  3. Combining the two parts above, we can write $(1,2)(3,4)(1,2,3) = (1,2)(3,4)(1,3)(1,2)$. This is a product of 4 transpositions, so the permutation is in $A_n$.
  4. The identity is a product of zero transpositions, which is even.
  5. We can write $(1,4,3,5)=(1,5)(1,3)(1,4)$ and also $(2,3,1,4,7,6)=(2,6)(2,7)(2,4)(2,1)(2,3)$. This means that $$(1,4,3,5)(2,3,1,4,7,6) = (1,5)(1,3)(1,4)(2,6)(2,7)(2,4)(2,1)(2,3), $$ which is the product of 8 transpositions.