TITLE: Do these two permutations generate $A_n$?
QUESTION [7 upvotes]: Let $n$ be odd  and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?

REPLY [5 votes]: $\newcommand{\Set}[1]{\left\{ #1 \right\}}$$\newcommand{\Span}[1]{\left\langle #1 \right\rangle}$No.
$$
\Span{(1,2,3,4,5,6,\dots,35), (1,6,11)}
$$
is not the alternating group on $35$ letters.
The reason is that it is imprimitive. A block is
$$
\Set{1,6,11,16,21,26,31}
$$