Exercise 1.7.8 ($S_A$ acts on $\mathscr{P}_k(A)$)

Proof. Note that if $X=\{a_1,a_2,\dots ,a_k\}\in B$, then $\sigma (a_1),\sigma (a_2),\dots ,\sigma (a_k)$ are distinct (because $\sigma$ is injective), therefore $\sigma \cdot X\in B$.

(a)

If $\sigma ,\tau$ are permutations in $S_A$, and $X=\{a_1,a_2,\dots ,a_k\}\in B$, then

• $$\begin{array}{llll}\hfill \sigma \cdot (\tau \cdot X)& =\sigma \cdot (\tau \cdot \{a_1,a_2,\dots ,a_k\})& \hfill & \\ \hfill & =\sigma \cdot \{\tau (a_1),\tau (a_2),\dots ,\tau (a_k)\}& \hfill & \\ \hfill & =\{\sigma (\tau (a_1)),\sigma (\tau (a_2)),\dots ,\sigma (\tau (a_k))\}& \hfill & \\ \hfill & =(\sigma \circ \tau )\cdot \{a_1,a_2,\dots ,a_k\}& \hfill & \\ \hfill & =(\sigma \circ \tau )\cdot X,& \hfill & \end{array}$$

• If $e={\mathrm{Id}}_A$, then

  $$\begin{array}{llll}\hfill e\cdot X& ={\mathrm{Id}}_A\cdot \{a_1,a_2,\dots ,a_k\}& \hfill & \\ \hfill & =\{{\mathrm{Id}}_A(a_1),{\mathrm{Id}}_A(a_2),\dots ,{\mathrm{Id}}_A(a_k)\}& \hfill & \\ \hfill & =\{a_1,a_2,\dots ,a_k\}& \hfill & \\ \hfill & =X.& \hfill & \end{array}$$

Therefore $S_A$ acts on the set $B$ by $\sigma \cdot \{a_1,\dots ,a_k\}=\{\sigma (a_1),\dots ,\sigma (a_k)\}$.

(b)

For instance $$\begin{array}{llll}\hfill (12)\cdot \{1,2\}& =\{2,1\}=\{1,2\},& \hfill & \\ \hfill (12)\cdot \{1,3\}& =\{2,3\},& \hfill & \\ \hfill (12)\cdot \{1,4\}& =\{2,4\},& \hfill & \\ \hfill (12)\cdot \{2,3\}& =\{1,3\},& \hfill & \\ \hfill (12)\cdot \{2,4\}& =\{1,4\},& \hfill & \\ \hfill (12)\cdot \{3,4\}& =\{3,4\},& \hfill & \end{array}$$

As expected, $(12)$ induces a permutation ${\phi }_{(12)}$ on $B={𝒫}_2([[1,4])$. Similarly

$$\begin{array}{llll}\hfill (123)\cdot \{1,2\}& =\{2,3\}& \hfill & \\ \hfill (123)\cdot \{1,3\}& =\{1,2\},& \hfill & \\ \hfill (123)\cdot \{1,4\}& =\{2,4\},& \hfill & \\ \hfill (123)\cdot \{2,3\}& =\{1,3\},& \hfill & \\ \hfill (123)\cdot \{2,4\}& =\{3,4\},& \hfill & \\ \hfill (123)\cdot \{3,4\}& =\{1,4\}.& \hfill & \end{array}$$

So $(123)$ induces also a permutation on $B={𝒫}_2([[1,4])$. □