Tell me if I got you right:

Well,

is given to be a Sylow p-subgroup of

. I also know that each conjugate of

is also a Sylow p-subgroup. Now

for all

because

. Since

, then

must be the only Sylow p-subgroup of

. Since

is a Sylow p-subgroup of

(which is also the only Sylow p-subgroup of

), then

must also be the only Sylow p-subgroup of

. Therefore,

.

With this proof, however, I used the fact that if

then

is unique. But the notes I have only guarantees the other way around: that if

is the only Sylow p-subgroup of

, then

. Is this statement a really biconditional? Because if it is, I would have to show it first.