Find two different $5$-Sylow subgroups in $S_{16}$.
Hint: use group multiplication.
Any hints?
Find two different $5$-Sylow subgroups in $S_{16}$.
Hint: use group multiplication.
Any hints?
What does a permutation of order 5 look like?
Can you give a condition when two permutations of order 5 commute?
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 $S_{16}$.