If you'd prefer to log in through Facebook instead of with an email address, just go to https://lets.paybygroup.com/login and click "Connect with Facebook." If it's your first time on Pay By Group, this will take you to a screen that asks for your Facebook login information. Once you approve the application that lets you sign in, you'll be brought to your group's dashboard.
Pay By Group will never post anything on your wall without your explicit permission.
NOTE: If you're already on the dashboard for the purchase you want to join, you can find the "Connect with Facebook" link below:
If you already have a Pay By Group account, simply log in as usual, then click on your name in the upper right of the screen to reveal a dropdown menu. Then click My Account.
Then, click on Link your Facebook account:
Then, log in to Facebook as you normally would. That's it!