Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 12)

887] 
ON THE THEORY OF GROUPS. 
641 
only 
from 
)lour 
that 
with 
agon 
from 
ides 
a 
rung 
of 
>f a 
of 
liar, 
the 
or 
IBG, 
ases, 
issed 
that 
m a 
the 
write 
arse 
, is 
so a 
lint; 
feba, 
may 
not 
ading 
the 
we 
Ferent 
these 
that 
from 
nitial 
every 
are 
any 
one initial point is a circuit from every other initial point; and we express this by 
saying that every route is of independent effect: this explains the meaning of the 
foregoing statement of the condition which is to be satisfied by a colourgroup. 
It is at once evident that a colourgroup, qua figure where each route is of 
independent effect, furnishes a graphical representation of the substitution-group and 
gives the square by which we define such group. For, in the colourgroup of a points, 
we have the route from a point to itself and the routes to each of the other 
(a — 1) points, in all « non-equivalent routes; and if starting from a given arrangement, 
say abed ..., of the a points, we go by one of these routes from the several points 
a, b, c, d, ... successively, we obtain a different arrangement of these points. Observe 
that this is so; the same point cannot occur twice, for if it did, there would be a 
route leading from two different points b, f to one and the same point x, or the 
reverse route from x would lead to two different points b, f The route from a 
point to itself which leaves each point unaltered, and thus gives the primitive 
arrangement abed ..., may be called the route 1. Taking this route and the other 
(a — 1) routes successively, we obtain a different arrangements of the points, or say a 
square, each line of. which is a different arrangement of the points. And not only 
are the arrangements different, but we cannot have the same point twice in any 
column, for this would mean that there were two different routes leading from a 
point to one and the same point x; hence each column of the square will be an 
arrangement of the a points. We have thus the substitution-group of the a points 
or letters; the a routes, or say the route 1 and the other (a — 1) routes, are the 
substitutions of the group. 
The complete figure is called the colourgroup. As already mentioned, the lines 
of any colour form either a single polygon or two or more polygons each of the 
same number of points. The number of lines of a given colour is thus = a, or when 
the polygons are digons (which implies a even), the number is =-|a. The number 
of colours is thus = (a — 1) at least, and = (a — 1) at most. A general description 
of the figure may be given as in the annexed Table. Thus, for the group 6B, we 
have 
R. 2 3gons = 6 
B, G, Y. (3 2gons) 3 = 9 
i|; 
we have the red lines forming two trigons, 6 lines, and the blue, green and yellow 
lines each forming three digons, together 3x3, =9 lines, in all 15, = 6.5 lines. 
Such description, however, does not indicate the currencies, and it is thus insufficient 
for the determination of the figure. But the figure is completely determined by 
means of the substitutions as given in the outside column of the square; thus 
R= (abc) (dfe) shows that the red lines form the two triangles abc, dfe with these 
currencies, G = (ad) (be) (cf), that the green lines form the three digons ad, be, ef, 
and so for the other two colours B and Y. 
The line of a colour may be spoken of as a colour, and the lines of a colour 
or of two or more colours as a colourset. The colourset either does not connect 
C. XII. 81
	        
Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.