Exercise 3.2.15 (If $G_i$ be the stabilizer of $i$ in $S_n$, then $G_i \simeq S_{n-1}$)

Proof. Consider set $S_X$ of permutations of $X=[[1,n]]\setminus \{i\}$. Since $|X|=n-1$, $S_X\simeq S_{n-1}$.

Let

so $\tau$ is the restriction of $\sigma$ to $X=[[1,n]]\setminus \{i\}$. Since $i$ is fixed by $\sigma$, $\tau$ is a permutation of $X$.

Then

• $\phi$ is a homomorphism: If ${\sigma }_1,{\sigma }_2\in G_i$, for all $j\in X=[[1,n]]\setminus \{i\}$,

  $$\phi ({\sigma }_1{\sigma }_2)(j)=({\sigma }_1{\sigma }_2)(j)={\sigma }_1({\sigma }_2(j))=\phi ({\sigma }_1)(\phi ({\sigma }_2)(j))=[\phi ({\sigma }_1)\phi ({\sigma }_2)](j),$$

  so $\phi ({\sigma }_1{\sigma }_2)=\phi ({\sigma }_1)\phi ({\sigma }_2)$.

• $\phi$ is injective: If $\sigma \in \ker <!--\; FUNCTION\; APPLICATION\; -->(\phi )$, then $\sigma (j)=j$ for all $j\ne i$, and since $\sigma \in G_i$, $\sigma (i)=i$, so $\sigma ={\mathrm{id}}_{[[1,n]]}$. Then $\ker <!--\; FUNCTION\; APPLICATION\; -->(\phi )={\mathrm{id}}_{[[1,n]]}$, os $\phi$ is injective.

• $\phi$ is surjective: Let $\tau \in S_X$. Put

  $$\sigma \{\begin{array}{ccc}\hfill [[1,n]]\hfill & \hfill \to \hfill & [[1,n]]\hfill \\ \hfill j\hfill & \hfill \mapsto \hfill & \sigma (j)=\{\begin{array}{cc}\tau (j)\hfill & \text{if }j\ne i,\hfill \\ i\hfill & \text{if }j=i.\hfill \end{array}\hfill \end{array}$$

  Then $\sigma \in S_n$ and by definition of $\sigma$, $\sigma (i)=i$, so $\sigma \in G_i$, and the restriction of $\sigma$ to $X$ is $\tau$, so $\phi (\sigma )=\tau$.

Therefore $\phi$ is an isomorphism, and

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