Alternative proof about the order of the alternating group?

Question

It is known that the order of the alternating group $A_n$ of order $n$ is $\frac{n!}{2}$. In Herstein's Abstract Algebra, it is proved by the First Homomorphism Theorem. I tried to find an alternative proof which needs not using the group homomorphism. I think the following rules may be helpful:

1. The product of two even permutations is even.
2. The product of two odd permutations is even.
3. The product of an even permutation by an odd one (or of an odd one by an even one) is odd.

Intuitively, since the product of an odd(resp. even) permutation and a 1-cycle is even(resp. odd), a half of all the permutations should be even. Then we get $\frac{n!}{2}$.

What's more, the theorem mentioned in this question may be related. I don't know if one can turn the argument above into a proof. So here is my question:

Does anybody know other proofs about the order of $A_n$?

Answer 1

Let $\text{Odd}_n$ denote the set of odd permutations in $S_n$. Fix an element $\alpha \in \text{Odd}_n$, and define a function $f\colon A_n \to \text{Odd}_n$ by $$ f(\sigma) \;=\; \alpha\sigma. $$ I claim that $f$ is a bijection.

To prove that $f$ is one-to-one, suppose that $f(\sigma) = f(\sigma')$ for some $\sigma,\sigma' \in A_n$. Then $\alpha\sigma = \alpha\sigma'$, and therefore $\sigma=\sigma'$.

To prove that $f$ is onto, let $\beta \in \text{Odd}_n$. Then $\alpha^{-1}\beta$ is an even permutation and $f(\alpha^{-1}\beta) = \beta$, which proves that $f$ is onto.

We conclude that $|A_n| = |\text{Odd}_n|$, and therefore $|A_n| = |S_n|/2$.

Answer 2

Clearly, we need to assume $n\geq 2$. The permutation $(12)$ is odd. The mapping $f:S_n\rightarrow S_n$, $f(\sigma)=(12)\circ\sigma$ is inverse to itself, hence bijective. If $\sigma$ is even, then $(12)\circ\sigma$ is odd. If $\tau$ is odd, then $(12)\circ \tau$ is even, and $\tau=(12)\circ(12)\circ\tau$. Hence $f(A_n)$ is the set of odd permutations. Since $f$ is bijective, there are as many even permuations as there are odd ones, i.e. there are $\frac{n!}{2}$ even and $\frac{n!}{2}$ odd permutations.

Answer 3

The following is, in my opinion, a nice exercise for students - you might ask them to use what they know about determinant to show that the number of odd permutations is the same as the number of even permutations. (But it seems to be more complicated than the proofs provided in other answers to this question.)

Let us consider the following determinant of the $n\times n$ matrix where each entry is $1$ $$E=\begin{vmatrix} 1 & 1 & \ldots & 1 \\ 1 & 1 & \ldots & 1 \\ 1 & 1 & \ldots & 1 \\ 1 & 1 & \ldots & 1 \end{vmatrix} $$ where $n\ge 2$.

Recall that determinant of a matrix $A$ is $$\operatorname{det}A=\sum_{\varphi\in S_n} (-1)^{i(\varphi)} a_{1\varphi(1)}a_{2\varphi(2)}\ldots a_{n\varphi(n)},$$ where $i(\varphi)$ denotes the number of inversions of $\varphi$.

We know that $(-1)^{i(\varphi)}$ is $+1$ for even permutation and $-1$ for odd permutation.

For the matrix $E$ we get $$\operatorname{det}E=\sum_{\varphi\in S_n} (-1)^{i(\varphi)} = \sum_{\varphi\in A_n} 1 + \sum_{\varphi\in S_n\setminus A_n} (-1) = |A_n|-(|S_n|-|A_n|)=2|A_n|-|S_n|.$$ But we know that $\operatorname{det} E=0$, since the rows are linearly dependent. So we get $2|A_n|-|S_n|=0$ and $$|A_n|=\frac{|S_n|}2.$$