There are two ways to create a group library, but both require that you have a Zotero online account.
When you create your group, the name you choose will also be the URL of the group library. Perhaps more importantly, you will need to select the membership type:
Once your group is created, you can invite members by email or Zotero username
Once a group has been created, you can edit three sets of settings:
Any group that you are the listed owner of counts towards your Zotero online storage limit. As a Dartmouth associated Zotero user, this amount is unlimited! You can leverage this by assigning group library ownership to Dartmouth users when collaborating with researchers outside of Dartmouth!
You can change group ownership permissions by logging in to your account at Zotero.org. Select Groups → Manage Library → Group Settings to update the ownership. Note that only current owners can reassign ownership, and ownership must be accepted by the new owner before the transfer is completed.