Hi Brian, this group button appeared because you (or a sync) created a group on your iPhone. To manage group of contacts (create or delete), we have developped a free app : Easy Group.
If you want to remove this Group button :
1- Download Easy Group : http://itunes.apple.com/us/app/easy-group/id461469079?mt=8&uo=4
2- Launch Easy Group, all your groups will appear
3- Tap "Edit"
4- Tap on red cross next to your unwanted groups.
Or you can simply use this group feature to classify your contacts, send group emails / texts.
Hope this helps.
Easy Group Community