Group type class with apply method.

Group.scala

object Group {
  def apply[T: Group]: Group[T] = implicitly[Group[T]]
  // Instances of Group[T]...
}

PairGroup.scala

implicit def pairGroup[T1: Group, T2: Group]: Group[(T1, T2)] = 
  new Group[(T1, T2)] {
    val zero = (Group[T1].zero, Group[T2].zero)
  
    def plus(x: (T1, T2), y: (T1, T2)): (T1, T2) = (x, y) match {
      case ((x1, x2), (y1, y2)) => 
        (Group[T1].plus(x1, y1), Group[T2].plus(x2, y2))
    }
  
    def inverse(x: (T1, T2)) = x match {
      case (x1, x2) => (Group[T1].inverse(x1), Group[T2].inverse(x2))
    }
  }

sum.scala

def sum[T: Group](elems: Seq[T]): T =
  elems.foldLeft(Group[T].zero)(Group[T].plus)
 
def sumNonEmpty[T: Group](elems: Seq[T]): Option[T] = 
  if (elems.isEmpty) None
  else Some(sum(elems))
 
def sumDifference[T: Group](elems1: Seq[Int], elems2: Seq[Int]): Int = 
  Group[T].minus(sum(elems1), sum(elems2))