Sylow $5$-subgroup in $S_{16}$ 
Find two different $5$-Sylow subgroups in $S_{16}$.
Hint: use group multiplication.  

Any hints?  

Since $16!$ is divisible by $5^3$, but not $5^4$, the $5$-Sylow subgroups of $S_{16}$ are the subgroups of order $5^3$.
An example is
$$\langle (1\ 2\ 3\ 4\ 5), (6\ 7\ 8\ 9\ 10), (11\ 12\ 13\ 14\ 15)\rangle.$$
(Since any pair of generators commutes, the group has isomorphism type $\mathbb Z/5\mathbb Z\times\mathbb Z/5\mathbb Z\times\mathbb Z/5\mathbb Z$ and in particular order $5^3$.)
Another example is
$$\langle (1\ 2\ 3\ 4\ 5), (6\ 7\ 8\ 9\ 10), (11\ 12\ 13\ 14\ 16)\rangle.$$
Both examples have a unique common fixed point of all elements.
In the first example, it is $16$, and in the second example it is $15$.
So the examples give two different $5$-Sylow subgroups of $S_{16}$.
