"Invitation only" Groups and default privacy settings change

Could anyone tell me if there is a way to make groups "invitation only".  Maybe there is some combination of the privacy settings that would make this work?  I think there are mods for this, but I am broke as usual, so if someone knows of a way I would really appreciate it.  Also, a related question.  Is there a way to change the default privacy setting for groups?  It is no where in the control panel but it must be possible with some code mods. 

It is for this site: http://VirtualBrainTrust.org.   Private groups are really important there, and I need this as soon as possible.  I could really use some volunteer help on that site by the way.  It is for a really good cause - you can read the basic concept of what I am trying to do here: http://goo.gl/HxDGE