Exercise 12, § 5

(a) Show that every permutation $\sigma \in \mathfrak{A}_n$ is a product of cycles of order 3 (which are not in general component cycles of $\sigma$). (Prove it for a product of two transpositions and use § 5, no. 7, Proposition 9.)

(b) If $a_1, \ldots, a_p$ are $p$ distinct elements of $\{1, n\}$, let $(a_1a_2\ldots a_p)$ denote the cycle of order $p$ whose support is $\{a_1, a_2, \ldots, a_p\}$ and which maps $a_i$ to $a_{i+1}$ for $1 \leq i \leq p-1$ and $a_p$ to $a_1$. Show that $\mathfrak{A}_n$ is generated by the $n-2$ cycles $(1\ 2\ 3), (1\ 2\ 4), \ldots, (1\ 2\ n)$. (Use (a).)

(c) Deduce that, if $n$ is odd, $\mathfrak{A}_n$ is generated by $(1\ 2\ 3)$ and $(1\ 2\ \ldots\ n)$ and, if $n$ is even, by $(1\ 2\ 3)$ and $(2\ 3\ \ldots\ n)$.

(d) Show that, if a normal subgroup of $\mathfrak{A}_n$ contains a cycle of order 3, it is identical with $\mathfrak{A}_n$ (prove that such a subgroup contains all the cycles $(1\ 2\ k)$, $3 \leq k \leq n$).