II. — Examples of Groups 



BY ELLERY WILLIAMS DAVIS 



The notion of a group is so simple and so useful an aid to the 

 logical memory that it can well be brought into even rather 

 elementary mathematics. For example, the operations of taking 

 the complement and taking the supplement of an angle generate 

 a group of order 8. To see this, 



let s mean supplement-of, 



and c mean complement-of; 



so that sc means supplement-of-complement-of. 



Then, 



if a is any angle 



we have Sa=Tr — a, ca=ir/2 — a. 



£ja=a — 7r/2, sca.=a-\-irj2 

 SCSa=2>' n '/ 2 — a > CSCa=: — a 

 CSCSa=:a — 7T, SCSc~a=a-{-Tr 



Two angles like a±7r differing by 2ir, or a complete revolution, 

 we regard as the same angle. 



It will be found that either ^ or c performed upon any one of 

 the above seven angles will produce another of them, or a. Thus, 

 we have the group of operations 



Gs^j.?, c\=\i, s, c, sc, cs, scs, esc, scsc\, 



such that the performance of any two in succession is the same 

 as the performance of some single operation of the group. We 

 call this single operation the product of the other two and con- 

 struct the multiplication table below. 



University Studies, Vol. IV, No. 3, July, 1904. 



231 



