Applications can be approved for groups that are set to: Private / apply to join.
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As the creator or the group or the Group Admin, you have the rights to approve or decline an application.

First, go to the group's detail page.
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On the detail page, click the button, [Number] applications to review.
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If the group required a user to apply but to not fill out an application form, you will see the buttons to approve or decline immediately. Click either Approve or Decline.

If the group required a user to apply and fill out an application form, click View Application to see the form submission. Then, on the form's result window, choose either Approve or Decline.