Sending alerts assigned to SharePoint Group is not supported by default in SharePoint 2010
But after implementing the following functionality:
- activate SharePoint Groups for selection in people editor control
- expand a group entity to individual user entities in the text box
it will become possible to send alerts to individual users from these groups
How it works
In Alert subscription page (New or Edit forms) type or select in People picker dialog box the SharePoint Group name

After clicking “Check names” button or OK button in People picker the SharePoint group will be expanded to individual users in the text box as shown below

For implementation details follow this article