Slowly but surely, Facebook for Android is getting to be more and more like the kind of app it should be instead of being a bare bones app. The latest update, version 1.5, has brought push notifications, chat capabilities, and bug fixes. It looks like the push notifications are for notifications you got before: friend requests, inbox messages, and events. And if you have chat turned off on the website, you will have to manually turn it off in the app. Enjoy!
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Source: AndroidAndMe