I would really appreciate someone's help if possible. I have set up a site with a series of sub sites, which I thought was working fine until I asked someone I had set up as an Owner of a sub site to do a test and try to add a new user. Although they have full control they only get the option to add a new group. They are in the Members group of the parent site and the permissions have been broken. I wondered if this had anything to do with it? I'm clutching at straws a bit here ... Any ideas, would be really helpful ... Thanks
Exactly, as in the previous answers, groups are managed only through their owners, not through the site owners. This was weird for me at first, too.
However, to work around that: Add the site owners as the owners for the newly created groups!
The above sugestions would work but would create a dangerous security vulnerability since now any member of the group would be able to add new members.
What you want to do is to create a new group that contains the various adminstrators of your site and asign this group as the owner of the group. This way any administrator can add new members but only administrators have this ability.
The ability to add new users to the group is generally controlled by "Group Settings":
As a site collection admin, select the group and select Settings > Group Settings to access it.
You need to change the Owner Option of the Group Owner.(In your case the SharePoint group to which you want to give access to add user to this group).
It doesnt matter if they have a full rights or not on this site they can still add users to this group if they are owner of this group.