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Re-reading POA I noticed that when the dementor comes onto the train and Lupin gets rid of it, Hermione doesn't know what it is, but surely after Dumbledore had mentioned they would be stationed at the entrances and exits, I would've assumed that Hermione would have gone to research about them.
Dumbledore mentions that the Dementors will be stationed outside the school grounds at the Start of Term Feast, immediately following the incident concerning the Dementor on the Hogwart's Express. So your time-line is slightly
skewed regarding whether or not she could have done research.
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I know she may have not had time, but she must have had some spare time at one point to spend ten minutes in the library to research them? I always wondered why she didn't. Because at the Quidditch match when they arrive on the pitch, she says in the Hospital Wing [just from memory] 'shot some silver stuff at them.'
There is an aspect to this that can be considered, but it takes some speculation. Naturally we are comfortable with speculation concerning the Potterverse considering most of us spent the entire era theorizing and dreaming up scenario after scenario to explain the mysteries and lingering questions. We have to decide whether we think the Ministry would keep the topic of the Dementors classified or not. We know that everyone knows about the Dementors, and they fear them, but do they really know that much about them? Would the Ministry allow details on how to evade/defeat Dementors (the patronus charm)? Would Fudge utilize the Dementors as personal guards if he knew it was easily obtained knowledge that repelled them? Perhaps any record of the Dementors, besides a basic description, had been magically omitted from Hogwarts Library seeing as they were guarding the castle. These are notions we need to take into consideration and decide which is most plausible.