GROUPS. In mathematics one frequently encounters sets of operations having the property that when two operations are carried out in succession the result is one that could be reached by a single operation of the set. Such a set of operations is called a closed set, and the property of closure is called the group property. A simple illustration is given by the four numbers, I, — 'I— I, and —'I— I of ordinary algebra where the effect of multiplying in succession any number by two of the set is the same as multiplying by a definite third. A formal definition of a group is given later under abstract groups.