In this BPOS support video, we'll learn how to troubleshoot permissions in SharePoint Online.
"SharePoint Online is one of the services included in Microsoft Online Services. SharePoint Online comes with permission control that allows you to determine the level of visibility for users over your company's SharePoint sites. Permissions can be set up per user or on a group basis."
The video below walks us through an overview of SharePoint Permissions, and the following common permission based troubleshooting scenarios: Permission Inheritance; Can't Add New User; User Not Receiving Alerts.