6.3: Alternating Groups
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 \(\PageIndex{4}\)
\(|A_n|= \dfrac{(n!)}{2} \text{.}\)