Are these conditions enough to specify a unique number? (counting sylow p-subgroups) Sylow's third theorem gives the two facts that the number of sylow p-subgroups $n_p$ of a group $G$, whose order we can write as $|G| = p^rm$ such that $p\not |\ m$ will satisfy both $n_p | m$ and $n_p \equiv 1\ (\mod\ p)$. I'm just wondering if this is enough to specify $n_p$ uniquely, or if there may still be ambiguity.
If you're wondering why I'm asking what might be such a basic question; I know next to nothing about number theory!

No, it's not enough. Take $|G| = 12 = 2^2 \cdot 3$, for example. Then $n_2 \equiv 1 \pmod{2}$ and $n_2 | 3$ gives that $n_2 \in \{1, 3\}$.
(In fact, there's no reason to take $2^2 \cdot 3$; $|G| = 6$ works just as well).
