6.3: Alternating Groups

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Note that every \(k\)-cycle \((a_1a_2\ldots a_k)\in S_n\) can be written as a product of (not necessarily disjoint) transpositions:

\begin{equation*} (a_1a_2\ldots a_k)=(a_1a_k)(a_1a_{k-1})\cdots(a_1a_3)(a_1a_2). \end{equation*}

We, therefore, have the following theorem.

Theorem \(\PageIndex{1}\)

Every permutation in \(S_n\) can be written as a product of transpositions.

Definition: Even and Odd

We say that a permutation in \(S_n\) is even [resp., odd] if it can be written as a product of an even [resp., odd] number of transpositions.

Theorem \(\PageIndex{2}\)

Every permutation in \(S_n\) is even or odd, but not both.

Proof: We already know that every permutation in \(S_n\) is a product of transpositions, so must be even or odd. For proof that no permutation is both even and odd, see, for instance, Proof 1 or 2 of Theorem 9.15 on p. 91 in [1].

Lemma \(\PageIndex{1}\)

For each \(2\leq k\leq n\text{,}\) then a \(k\)-cycle is even if \(k\) is odd, and odd if \(k\) is even.

The proof of this is left as an exercise for the reader.

Example \(\PageIndex{1}\)

In \(S_3\text{,}\) the permutations \(e\text{,}\) \((123)=(13)(12)\text{,}\) and \((132)=(12)(13)\) are even, while the permutations \((12)\text{,}\) \((13)\text{,}\) and \((23)\) are odd.

Example \(\PageIndex{2}\)

List all of the even [resp., odd] permutations in \(S_4\text{.}\)

We have the following theorem, whose proof is left as an exercise for the reader.

Theorem \(\PageIndex{3}\)

The set of all even permutations in \(S_n\) is a subgroup of \(S_n\text{.}\)

Definition: Alternating Group

The alternating group on \(n\) letters is the subgroup \(A_n\) of \(S_n\) consisting of all of the even permutations in \(S_n\text{.}\)

We end with this theorem, whose proof can be found on p. 93 of [1].

Theorem \(\PageIndex{4}\)

\(|A_n|= \dfrac{(n!)}{2} \text{.}\)