The group based fields in the Fields drop-down exist to work with your new group.
There are five different group related fields you can add:
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Group field = this is added automatically - it's a toggle that you use to mark a group node type as a group
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Group roles and permissions field = this field provides the ability to override the default roles and permission sets for your group.
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Group visibility field = this field will provide the ability to make a group public or private.
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Group register field = this field will provide a link for users to request to register with the group.
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Groups views field = this field is not well understood at this time
The way to add each of these fields is the same, but, depending on the field you choose, there may be different configuration options within the field settings at Admin menu > Structure > Content types > Your Group Node Type > Manage Fields. Please note that, for most of these fields, the default settings work just fine.
As you add the fields, though, they will be visible on display. You may want to go into Admin menu > Structure > Content types > Your Group Node Type > Manage Display and hide the fields.
OSU Drupal 7 only has the Group field included and preconfigured.