694] 
DESIDERATA AND SUGGESTIONS. 
403 
a/3 2 (a 2 = 1, /3 s = 1); viz. in the first of these a/3 =/3a, while in the other of them (that 
mentioned above) we have a/3 = /3 2 a, a/3 2 = /3a. 
But although the theory as above stated is a general one, including as a 
particular case the theory of substitutions, yet the general problem of finding all the 
groups of a given order n, is really identical with the apparently less general problem 
of finding all the groups of the same order n, which can be formed with the substitu 
tions upon n letters; in fact, referring to the diagram, it appears that 1, a, /3, 7, 8, e 
may be regarded as substitutions performed upon the six letters 1, a, /3, 7, 8, e, 
viz. 1 is the substitution unity which leaves the order unaltered, a the substitution 
which changes la/3y8e into al<y/3e8, and so for /3, 7, 8, e. This, however, does not in 
any wise show that the best or the easiest mode of treating the general problem is thus 
to regard it as a problem of substitutions: and it seems clear that the better course 
is to consider the general problem in itself, and to deduce from it the theory of 
groups of substitutions. 
Cambridge, 26th November, 1877. 
No. 2. The theory of groups; graphical representation. 
[From the American Journal of Mathematics, t. 1. (1878), pp. 174—176.] 
In regard to a substitution-group of the order n upon the same number of letters, 
I omitted to mention the important theorem that every substitution is regular (that 
is, either cyclical or composed of a number of cycles each of them of the same order). 
Thus, in the group of 6 given in No. 1, writing a, b, c, d, e, f in place of 1, a, 
/3, 7, 8, e, the substitutions of the group are 1, ace.bfd, aec.bdf, ab.cd.ef ad.be.cf, 
af. be. de. 
Let the letters be represented by points; a change a into b will be represented 
by a directed line (line with an arrow) joining the two points; and therefore a cycle 
abc, that is, a into b, b into c, c into a, by the three sides of the trilateral abc, 
with the three arrows pointing accordingly, and similarly for the cycles abed, &c.: 
the cycle ab means a into b, b into a, and we have here the line ab with a two- 
headed arrow pointing both ways; such a line may be regarded as a bilateral. A 
substitution is thus represented by a multilateral or system of multilaterals, each side 
with its arrow; and in the case of a regular substitution the multilaterals (if more 
than one) have each of them the same number of sides. To represent two or more 
substitutions we require different colours, the multilaterals belonging to any one 
substitution being of the same colour. 
In order to represent a group we need to represent only independent substitutions 
thereof; that is, substitutions such that no one of them can be obtained from the 
others by compounding them together in any manner. I take as an example a group 
51—2