I would like to know, if there is any possibility to write an inner search engine into a Facebook group. Explained a little larger: there are groups in a FB account, in the group some topics to talk about. After a period, a new topic is introduced, which was already discussed (one of the general problems of the forums). Many forums solved this problem (like this) with an inner search engine, so when you open a new topic, it automatically looks through the possible matchings and suggests to the user some already opened and discussed topics. Is it possible to write an application which does this search on Facebook groups topics? Would it be possible to create a search engine which looks through the user's documents, posts, messages?
Yeah, in theory you should be able to pull something like that out.
You can get all posts of a group using FQL, for example here are all of the messages posted in a group feed:
SELECT post_id, actor_id, target_id, message FROM stream WHERE source_id="GROUP_ID"
You'll have to perform the text search yourself though, from the data that you get from facebook. Also, you'll need to get the "user_groups" permission for that.
Another thing is that you won't be able to change the actual UI of the group inside facebook, and so this search of yours won't be available for users who post a new message on the group feed.
Hope this is what you wanted.