I don't believe that's necessarily true, x. It would depend largely on the circumstances of course, but to make the blanket statement that all abusers are necessarily unbelievers. Given that our sanctification is a journey and not just a destination, I would think it possible that one could be abusive while still being a believer, though obviously a believer engaged in disobedience and in sore need of repentance.
As you can see, I posted immediately BEFORE you revising my statement a bit. It shows us to be in agreement.
A "walk" in darkness is evidence that one is not of the Father.