You can post, Amazon, affiliate links, and Facebook groups as long as the group is public. All you do is create the Amazon affiliate links and then go about sharing them in the Facebook group. When somebody clicks on your link in the Facebook group, you want megs a percentage of the sale when things are purchase through that link that you posted.
I would advise that you do not share any Amazon affiliate links in private groups. This is how I got my Amazon affiliate account terminated. The fact that the Amazon employees can’t go into the group openly to check on your legs is against their terms of service, so make sure that your Facebook group is set to public or else you’re going to break the Amazon terms of service.