Do you invite people to join groups you've made? I've been looking everywhere on my group profile to no avail.

go to their profile, click join group

EDIT: it's add to group actually
There's no such thing; there never was. Where I am going you cannot follow me now.
I think on your group profile there is an invite link, just add whoever you want to be on your list and from there you should be set unless no one likes your group. But I am not too sure about this.
#25 for top 100 UGer of 2009
UG's 2nd Funniest UGer and 3rd most likely to be a Serial Killer of 2009, 2nd of 2011