690] 
ON THE THEORY OF GROUPS. 
325 
1 
A set of symbols a, /3, 7,.., such that the product a/3 of each two of them 
(in each order, a/3 and /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 symbols (or terms) is = n, then the group is of the order n; 
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 ??th roots of unity. 
A group is defined by means of the laws of combinations 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 distinct 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 a), 
where observe that these conditions imply also a/3 2 =/3a. 
Or in the second mode, calling the symbols (1, a, /3, a/3, f3 2 , a/3 2 ) of the same 
group (1, a, /3, 7, 8, e), or, if we please, (a, b, c, d, e, /), the laws of combination 
are given by one or other of the square diagrams : 
1 a /3 7 8 € 
1 
1 
a 
/8 
y 
8 
£ 
a 
a 
1 
y 
/8 
€ 
8 
/? 
/8 
€ 
8 
a 
1 
y 
y 
y 
8 
c 
1 
a 
J8 
8 
8 
y 
1 
e 
/3 
a 
e 
£ 
/8 
a 
8 
y 
1 
a 
b 
c 
d 
e 
f 
b 
a 
d 
c 
f 
e 
c 
f 
e 
b 
a 
d 
d 
e 
f 
a 
b 
c 
e 
d 
a 
f 
c 
b 
f 
c 
b 
e 
d 
a 
where, taking for greater symmetry the second form of the square, observe that the 
square is such that no letter occurs twice in the same line, or in the same column (or 
what is the same thing, each of the lines and of the columns contains all the letters). 
But this is not sufficient in order that the square may represent a group; the square 
must be such that the substitutions by means of which its several lines are derived 
from any line thereof are (in a different order) the same substitutions by which the 
lines are derived from a particular line, or say from the top line. These, in fact, are: