887]
ON THE THEORY OF GROUPS.
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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