402 DESIDERATA AND SUGGESTIONS. [694 
The functional symbols may be substitutions; a (x, y, z) = (y, z, x), the same letters 
in a different order: substitutions can be represented by the notation a — - ZX , the 
xyz 
substitution which changes xyz into yzx, or as products of cyclical substitutions, 
a = y ZX WU , = (xyz) (uw), the product of the cyclical interchanges x into y, y into z, 
xyz uw 
and z into x; and u into w, w into u. 
A set of symbols a, /3, 7, such that the product a/3 of each two of them (in 
each order, a/3 or /3a.), is a symbol of the set, is a group. It is easily seen that 1 
is a symbol of every group, and we may therefore give the definition in the form 
that a set of symbols, 1, a, /3, 7,... satisfying the foregoing condition is a group. 
When the number of the symbols (or terms) is = n, then the group is of the wth 
order; and each symbol a is such that a n = 1, so that a group of the order n is, 
in fact, a group of symbolical wth roots of unity. 
A group is defined by means of the laws of combination of its symbols: for the 
statement of these we may either (by the introduction of powers and products) 
diminish as much as may be the number of independent functional symbols, or else, 
using distinct letters for the several terms of the group, employ a square diagram 
as presently mentioned. 
Thus, in the first mode, a group is 1, /3, /3 2 , a, a/3, a/3 2 (a 2 =l, /3 3 =1, a/3 = /3 2 u); 
where observe that these conditions imply also a/3 2 — /3a. 
Or, in the second mode, calling the same group (1, a, /3, 7, 8, e), the laws of 
combination are given by the square diagram 
1 
a 
/5 
y 
8 
e 
1 
1 
a 
/? 
y 
8 
c 
a 
a 
1 
y 
/3 
€ 
8 
/3 
~co 
€ 
8 
a 
1 
y 
r 
y 
8 
€ 
1 
a 
/3 
8 
8 
y 
1 
€ 
/3 
a 
e 
€ 
/3 
a 
8 
y 
1 
for the symbols (1, a, /3, 7, 8, e) are in fact = (1, a, /3, a/3, /3 2 , a/3 2 ). 
The general problem is to find all the groups of a given order n; thus if n = 2, 
the only group is 1, a (a 2 = l); if n = 3, the only group is 1, a, a 2 (a 3 = l); if n= 4, the 
groups are 1, a, a 2 , a 3 (a 4 =l), and 1, a, /3, a/3 (a 2 =l, /3 2 = 1, a/3 =/3a)-, if n — 6, there 
are three groups, a group 1, a, a 2 , a 3 , a 4 , a 5 (a 6 = 1), and two groups 1, ¡3, (3 2 , a, a/3,