Nope, no member application process. Basically, you just PM the social group leader of the social group you want to join, and express your interest and perhaps a little bit about why you want to join their group.

If the social group is active, the member will PM you back and accept your invite. If they don't, they'll provide a reason for not doing so (maybe it's a special group setup just for their friends or something, but that would be rare).

DocJohn