Always ask yourself this:
Why would my customers want to do this?
So, in your case:
Why would my customers want to interact on this Facebook group? They need a reason, because otherwise they simply won't do it! So give them some sort of incentive. Let me tell you a quick story - I just bought a brand new car last week, and the dealership asked me to become a fan of their page on Facebook, and in return they filled my tires with nitrogen - which normally costs about 40$, I believe. But hey, I now find myself randomly checking their page for updates and things of that nature - so their plan worked.