Exercise 23, § 5

(a) Let $s$ be an element of order 2 in the group $\mathfrak{S}_n$ which is a product of $k$ transpositions $t_1, \ldots, t_k$ with mutually disjoint supports. Show that the centralizer $C_s$ of $s$ in $\mathfrak{S}_n$ is isomorphic to $\mathfrak{S}_{n-2k} \times A$, where $A$ admits a normal subgroup of order $2^k$ such that $A/B$ is isomorphic to $\mathfrak{S}_k$ (observe that every element of $C_s$ permutes the fixed points of $s$ and also the supports of $t_1, \ldots, t_k$).

(b) Show that, if $n = 5$ or $n \geq 7$, every element of order 2 in $\mathfrak{S}_n$ whose centralizer is isomorphic to that of a transposition is a transposition (compare the Jordan-Hölder quotients of these centralizers using Exercise 16). Deduce that every automorphism of $\mathfrak{S}_n$ permutes the transpositions.

(c) If $a, b$ are distinct elements of $\{1, n\}$, let $F_a$ (resp. $F_{ab}$) denote the stabilizer of $a$ (resp. the fixer of $\{a, b\}$) in $\mathfrak{S}_n$ and $(a\ b)$ the transposition $\tau_{a,b}$. Let $a, b, c, d$ be distinct elements of $\{1, n\}$. Show that, if $n \geq 5$, the group generated by $F_{ab}$ and $F_{cd}$ is $\mathfrak{S}_n$ (show that this group contains all the transpositions; for this use the fact that, if $x \in \{1, n\} - \{a, b, c, d\}$, then $(a\ c) = (c\ x)(a\ x)(c\ x)$). Show that the group generated by $F_{ab}$ and $F_{ac}$ is $F_a$ (use the same equation as in the above case).

(d) Show that an automorphism $\alpha$ of $\mathfrak{S}_n$ such that $\alpha(F_a) = F_a$ for all $a \in \{1, n\}$ is the identity (show that $\alpha$ maps every transposition to itself). Show that, if there exists $a \in \{1, n\}$ such that $\alpha(F_a) = F_a$, $\alpha$ is an inner automorphism (show that there exists an inner automorphism $\beta$ such that $\beta \circ \alpha$ maps each of the groups $F_b, b \in \{1, n\}$, to itself).

(e) Let $a, b$ be two distinct elements of $\{1, n\}$ and let $C_{ab}$ be the centralizer of $(a, b)$ in $\mathfrak{S}_n$. Show that, if $n \geq 5$, the derived group of $C_{ab}$ is of index 4 in $C_{ab}$ (use the fact that the derived group of $\mathfrak{S}_{n-2}$ is $\mathfrak{A}_{n-2}$). Show that $F_{ab}$ is the unique subgroup of $C_{ab}$ of index 2 which does not contain $(a\ b)$ and is not contained in $\mathfrak{A}_n$.

(f) Show that, if $n \geq 5$, an automorphism $\alpha$ of $\mathfrak{S}_n$ which leaves fixed a transposition $(a\ b)$ is inner. (Show by means of (e) that $\alpha(F_{ab}) = F_{ab}$; show equally that, if $c \notin \{a, b\}$, $\alpha(F_{ac})$ is either of the form $F_{ax}$, $x \notin \{a, b\}$, or of the form $F_{bx}$, $x \notin \{a, b\}$; in the first case conclude that $\alpha(F_a) = F_a$ and apply (d); reduce the second case to the first by multiplying $\alpha$ by the inner automorphism of $\mathfrak{S}_n$ defined by $(a\ b)$.)

(g) Deduce from (b) and (f) that every automorphism of $\mathfrak{S}_n$ is inner if $n = 5$ or $n \geq 7$.