Hello Nigel,
Thank you for the question.
As I've mentioned in my reply to Dan above, communities can form pretty much anywhere. And according to the definition in this article FB Groups would be a community, despite the fact that Facebook is a social network. Because you don't necessarily know everyone in the FB group. You can have many FB groups based on your interest. So it is definitely a community.
Social Network and Community are not mutually exclusive. We are part of 1 social network, but we are also part of many community. Strictly from a social anthroplogical view, there is a very important reason for us to have both of these social structures. These more indepth topics will be cover in a miniseries of article that I will post later.
The question you've ask is precisely why I decided to start this mini-series in the first place. The answer is more involved and not trivial. It has to do with the dynamics of how ties form and the differential tendency to engage with friends with different tie strength. So if you follow my blog, you will find the answer to your question: why FB groups are not very effective. Stay tuned!