There is something seriously wrong with your user permissions. Can you go to admin console, Permissions -> Social group permissions and check what permissions are given to All registered Users ?
"All registered Users" should not have Manage Groups permission.
Also, you may want to review space and system permissions as well
I follow your steps and removed the Manage Groups permission for all registered users.
This option let all the users see, edit and post content in every group, no matter if they are members.
Thank you for your help. Your tips were really usefull.