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Definition ($A_n$ - Alternating Group of Degree $n$)

Let $n\in\mathbb{N}$. Suppose that $\alpha\in S_n$.

  1. We say that $\alpha$ is a transposition if $\alpha$ is a cycle $(a,b)$ of length 2.
  2. We say that $\alpha$ is an even permutation if we can write $\alpha$ as the product of an even number of transpositions. Otherwise, we say that $\alpha$ is odd.
  3. The alternating group of degree $n$, written $A_n$, is the subset of $S_n$ of all even permutations.

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