I'm not completely sure the script will be faster than your current set up, but Set Field in a script can move data from field in the related table into a global field or into a field defined in the person table.
Set Field [Person::Yourfield ; Group::GroupName ] //this accesses the first related record.
If you have FileMaker 11, you could also use Set Variable to move the data into a variable which you can then place on this layout as merge text <<$$VariableNameHere>> to display the name.
I do have filemaker 11. So how would I setup the Set Variable to move the data into the new variable? Also, not sure what you mean by merge text?
Set Variable [$$GroupName ; value: Group:GroupName]
If you enter layout mode, you can then use the layout text tool to add this layout text to your layout:
If $$GroupName is empty, you'll see the above text when in browse mode. When you run the above script, you'll see the group name instead.