Ad the "connect with facebook" button on the register page

The file you'll need to edit is the /components/com_community/templates/<YOUR_TEMPLATE>/register.index.php As always, it's best to create a template override by copying that file to /templates/<YOUR_TEMPLATE>/html/com_community/register.index.php and editing there.

Once you are editing, you can add the tag anywhere. You should be able to see the full registration form in there with all the HTML. I could see the tag going at the top, bottom, or even on the side, but it would depend on how you want it to look. Hopefully, you can test adding it in that file and get something you like.

Best of luck,
Alex