Note on the Groups of Subtraction and Division, and on the Hyperbolic Functions

Extract

The objects of the present note are to exhibit an interesting property of some groups of subtraction and division, and to present a method of using such groups in the study of some elementary properties of the hyperbolic functions. The four finite groups of subtraction and division which transform every rational number into a rational number are of orders 4, 6, 8, and 12 respectively. With respect to each of these groups all the numbers, with the exception of two more than the order of the group, are divided into sets of conjugates, each set containing as many distinct numbers as the order of the group. The property to which we referred in the first sentence is that each of these sets contains two and only two negative numbers whenever the set is composed of real numbers and the number from which we subtract is positive.