Exercise 3.5.5 ($S_p = \langle \sigma, au angle$ where $\sigma$ is any transposition and $ au$ is any $p$-cycle)

Question

Show that if $p$ is prime, $S_p = \langle \sigma, 	au angle$ where $\sigma$ is any transposition and $	au$ is any $p$-cycle.

Richard Ganaye · Accepted Answer

\begin{proof} 
Let $\sigma$ be any transposition and $	au$ be any $p$-cycle. Then
$$	au = (a_1\ a_2\ \cdots\ a_p),$$
where $\{a_1,a_2,\ldots a_p\} = [\![1,p]\!]$ (see Exercise 1.3.10).
Therefore
$$\sigma = (a_i \ a_j) $$
for some integers $i,j$ such that $1 \leq i < j \leq p$. 

Then 
$$a_j = 	au^{j-i}(a_i).$$
Consider the permutation $\lambda = 	au^{j-i}$, so that $\lambda(a_i) = a_j$. We show first that $S_n = \langle \sigma, \lambda angle$.
Since $1 \leq i < j \leq p$, we obtain $1\leq j-i<p$, where $p$ is prime, therefore $\mathrm{g.c.d}(j-i, p) = 1$. By Exercise 1.3.11, this implies that $\lambda = 	au^{j-i}$ is also a $p$-cycle, and by Exercise 1.3.10, 
\begin{align*}
\lambda  &= (a_i \ \lambda(a_i) \ \lambda^2(a_i) \ \cdots \ \lambda^{p-1}(a_i)),\
\sigma &= (a_i\ \lambda(a_i)).
\end{align*}
Consider the permutation
$$\alpha = 
\begin{pmatrix}
1&2 &\cdots & p\
a_i & \lambda(a_i)& \cdots & \lambda^{p-1}(a_i)
\end{pmatrix}
$$
defined by $\alpha(k) = \lambda^{k-1}(a_i)$ (this makes sense because $a_i ,\lambda(a_i), \lambda^2(a_i),\ldots, \lambda^{p-1}(a_i)$ are distinct).

By the lemma proven in Exercise 4,
\begin{align*}
\sigma &= \alpha \, (1\ 2)\,  \alpha^{-1},\
\lambda &= \alpha\,  (1\ 2\ \cdots \ p)\,  \alpha^{-1}.
\end{align*}

Consider the inner automorphism $f = \gamma_{\alpha}$ defined by $f(\delta) =  \alpha\,  \delta \,  \alpha^{-1}$ for every permutation $\delta \in S_n$. Then $f((1\ 2)) = \sigma $ and $f((1\ 2\ \cdots \ p))=\lambda $.

By  Exercise 4, $S_p = \langle (1\ 2), (1\ 2\ \cdots \ p)angle$.

Let $\xi$ by any permutation in $S_n$. Since $f$ is an automorphism, there is some permutation $\zeta$ such that $\xi = \alpha \zeta \alpha^{-1}= f(\zeta)$.  Since $S_p = \langle (1\ 2), (1\ 2\ \cdots \ p)angle$, by Proposition 9,
$$\zeta = \sigma_1^{\varepsilon_1} \sigma_2^{\varepsilon_2}\cdots  \sigma_k^{\varepsilon_k},$$
where $\varepsilon_i \in \{-1,1\}$ and $\sigma_i \in \{(1\ 2), (1\ 2\ \cdots \ p)\}$ for $i \in [\![1,k]\!]$. Therefore
\begin{align*}
\xi &= f(\zeta)\
&= f(\sigma_1^{\varepsilon_1} \sigma_2^{\varepsilon_2}\cdots  \sigma_k^{\varepsilon_k})\
&= 	au_1^{\varepsilon_1} 	au_2^{\varepsilon_2} \cdots 	au_1^{\varepsilon_k},
\end{align*}
where $	au_i = f(\sigma_i )\in \{\sigma, \lambda\}$. This shows that
$$S_n = \langle \sigma, \lambda angle = \langle \sigma, 	au^{j-i} angle.$$
Since $\langle \sigma, 	au^{j-i} angle \subseteq \langle \sigma, 	au angle$, this shows that $S_n \subseteq \langle \sigma, 	au angle$, where $\langle \sigma, 	au angle \subseteq S_n$, so
$$S_n = \langle \sigma, 	au angle.$$

\bigskip
 If $p$ is prime, $S_p = \langle \sigma, 	au angle$ where $\sigma$ is any transposition and $	au$ is any $p$-cycle.
\end{proof}

Exercise 3.5.5 ($S_p = \langle \sigma, \tau \rangle$ where $\sigma$ is any transposition and $\tau$ is any $p$-cycle)

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Exercise 3.5.5 ($S_p = \langle \sigma, \tau \rangle$ where $\sigma$ is any transposition and $\tau$ is any $p$-cycle)

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