If you were creating a group user the purpose of this would be to add an email address that would email a group of people in your organisation. You cannot “assign” users to a group user.
@iamoat: To elaborate on Will’s answer, a Case or any other record type can be assigned to a specific user, NOT to a group that contains multiple users.
A “group user” is like any other user with the exception of not being able to login to the system. Its sole purpose is to contain multiple email addresses that need to be notified for something. Do not be confused by the term “group” in the user type. You might want to look into using a Workflow to accomplish what you want.