Where is Address Book in Mountain Lion?
Solution 1:
In the new Contacts Application for Mountain Lion there are no buttons for group management, only menu items.
Under the File Menu, there are New Group, New Group from Selection, and New Smart Group… menu items.
When you have a group selected, under the Edit menu there will be a Delete Group menu item available.
To be able to select groups, you need to be in the proper view under the View menu: Groups.