Why is the member's email not showing in the group member tab?

If the member's email is not showing in the group member tab it is because the email is not currently in the member record.  Click on the Member's Name and it will bring you to the Member Record where you can add the email. 

Add the Email and check Email Notifications check box, then click Save. 

Once this is done it will show in the Group Member Tab and the member can receive emails when emailing from the Group Module.