## Alt(n+2) contains a subgroup isomorphic to Sym(n)

Prove that $A_n$ contains a subgroup isomorphic to $S_{n-2}$ for all $n \geq 3$.

Let $n \geq 1$. We saw in a previous theorem that $S_n = \langle B \rangle$ where $B = \{ (i\ j) \ |\ 1 \leq i < j \leq n \}$.

Let $n \geq 1$, and let $\sigma = (n+1\ n+2)$. Define $\varphi : S_n \rightarrow A_{n+2}$ by $\varphi(\tau) = \tau$ if $\tau$ is even and $\varphi(\tau) = \tau\sigma$ if $\tau$ is odd.

1. $\varphi$ is a homomorphism: If $\alpha$ and $\beta$ are even, then $\varphi(\alpha\beta) = \alpha\beta = \varphi(\alpha)\varphi(\beta)$. If without loss of generality $\alpha$ is even and $\beta$ is odd, $\varphi(\alpha\beta) = \alpha\beta\sigma = \varphi(\alpha)\varphi(\beta)$. If $\alpha$ and $\beta$ are odd, then $\varphi(\alpha\beta) = \alpha\beta = \alpha\beta\sigma^2 = \alpha\sigma\beta\sigma = \varphi(\alpha)\varphi(\beta)$.
2. $\varphi$ is injective: Suppose $\alpha \in \mathsf{ker}\ \varphi$. Then since $1$ is even and $\varphi(\alpha) = 1$, we have $\alpha = 1$. So the kernel of $\varphi$ is trivial and $\varphi$ is injective.

So $\varphi$ embeds $S_n$ as a subgroup of $A_{n+2}$. Moreover, recall that $S_n = \langle B \rangle$ where $B = \{ (i\ j) \ |\ 1 \leq i < j \leq n \}$. So this copy of $S_n$ in $A_{n+2}$ is generated by $B\sigma$.

• rene  On September 23, 2011 at 12:46 am

Hi, just wondering why is step 3 necessary.
If you have your injective isomorphism, then phi(Sn) is a subgroup of A_n isomorphic to S_n-2, right?

• nbloomf  On September 23, 2011 at 10:04 am

Step 3 is indeed superfluous. At first I thought it was useful for giving a specific generating set for the copy of $S_n$ in $A_{n+2}$, but once we know that $\varphi$ is injective, we get the generating set $B\sigma$ for free.

Thanks!

• Carlos  On December 5, 2011 at 9:47 pm

Quick question: why can we commute \sigma and \beta when beta is odd?

• nbloomf  On December 5, 2011 at 10:00 pm

They are disjoint. That is, $\sigma$ fixes the elements $\beta$ moves and vice versa. It’s not too hard to show that disjoint permutations commute. (D&F discuss this on page 32.)