694] 
401 
694. 
DESIDERATA AND SUGGESTIONS. 
No. 1. The theory of groups. 
[From the American Journal of Mathematics, t. I. (1878), pp. 50—52.] 
Substitutions, and (in connexion therewith) groups, have been a good deal 
studied; but only a little has been done towards the solution of the general problem 
of groups. I give the theory so far as is necessary for the purpose of pointing out 
what appears to me to be wanting. 
Let a, /3,... be functional symbols, each operating upon one and the same number 
of letters and producing as its result the same number of functions of these letters ; 
for instance, a (x, y, z) = (X, Y, Z), where the capitals denote each of them a given 
function of (x, y, z). 
Such symbols are susceptible of repetition and of combination ; 
a?(x, y, z) = a{X, Y, Z), or /3a (x, y, z) = /3(X, Y, Z), 
= in each case three given functions of (x, y, z) ; and similarly for a 3 , a 2 /3, &c. 
The symbols are not in general commutative, a/3 not = /3a ; but they are as 
sociative, a/3.7 = a. /37, each = a/3y, which has thus a determinate signification. 
The associativeness of such symbols arises from the circumstance that the 
definitions of a, /3, 7, ... determine the meanings of a/3, ay, &c. : if a, /3, 7,... were 
quasi-quantitative symbols such as the quaternion imaginaries i, j, k, then a/3 and f3y 
might have by definition values 8 and e such that a/3.7 and a. /3y (= 8y and ae 
respectively) have unequal values. 
Unity as a functional symbol denotes that the letters are unaltered, l(x, y, z)=(x, y, z); 
whence la = al = a. 
C. x. 
51