Facebook blocking help centre link.
I found a problematic person, whom I had "blocked" was able to communicate on a group of which he is a member (I am administrator of the group). Turns out that
Once you block someone, that person can no longer be your friend on Facebook or interact with you (except within apps and games you both use and groups you are both a member of). [Facebook]
So what happens is that the person can't post on your profile but gets to post on groups he/she has joined but of which you are administrator.
Groups: You’ll be able to see groups that the blocked person created or belongs to, but you won’t see that person in the group member list. You also won’t see anything that person posts in the group.
Thus, someone wrote on FB:
I've "permanently" banned fans and they keep coming back over and over. I no longer see the comments they leave, but other fans do? Tell me what I need to do–as an administrator–to require people to get approval before they are accepted as fans. [Source]
What this means is that even as administrator you wont' be able to see the posts made by this person, but will know because others are responding!
This is a particular problem when people are posting SPAM. In that case use the SPAM option, not the block option.
The problem with the block option is that when you want to undo that, the list is ordered in a random manner so you can't remember whom you unblocked for spam. So always use the SPAM option.
Facebook ban permanently
As administrator you can PERMANENTLY ban a member. Go to the member list (Members > see all), then search for the member and click the cross button next to his name – that will give you the choice of banning him permanently.