Exercise 24, § 5

(a) Show that there exists a subgroup $H$ of $\mathfrak{S}_6$ which is isomorphic to $\mathfrak{S}_5$ and leaves fixed no element of $\{1, 6\}$ (let $\mathfrak{S}_5$ operate by inner automorphisms on the set of its subgroups of order 5, which has 6 elements).

(b) Show that there exists an automorphism $\sigma$ of $\mathfrak{S}_6$ which maps $(1\ 2)$ to $(1\ 2)(3\ 4)(5\ 6)$. (Let $\mathfrak{S}_6$ operate on $\mathfrak{S}_6/H$, where $H$ is chosen as above.)

(c) Show that the group of inner automorphisms of $\mathfrak{S}_6$ is of index 2 in the group of all automorphism of $\mathfrak{S}_6$. (Let $\alpha$ be an automorphism of $\mathfrak{S}_6$; show that $\alpha$ maps a transposition either to a transposition or to a conjugate of $(1\ 2)(3\ 4)(5\ 6)$. Using Exercise 23, show that in the first (resp. second) case $\alpha$ (resp. $\alpha \circ \sigma$) is an inner automorphism.)