Exercise 10, § 5

(a) Show that the elements $\tau_{12}\tau_{34}, \tau_{13}\tau_{24}, \tau_{14}\tau_{23}$ of the alternating group $\mathfrak{A}_4$ form with the identity element a commutative subgroup H of $\mathfrak{A}_4$. Show that H is normal in $\mathfrak{A}_4$ and in $\mathfrak{S}_4$ and that $\mathfrak{A}_4/H$ is cyclic of order 3.

(b) Show that the centralizer of H in $\mathfrak{S}_4$ is equal to H. Deduce that the mapping $s \mapsto (h \mapsto shs^{-1})$ of $\mathfrak{S}_4$ into Aut(H) defines when passing to the quotient an isomorphism of $\mathfrak{S}_4/H$ onto Aut(H) and that the latter group is isomorphic to $\mathfrak{S}_3$.

(c) Let K be a subgroup of H of order 2. Show that K is not normal in $\mathfrak{A}_4$.