690] 
ON THE THEORY OF GROUPS. 
329 
by the diagram, as presently appearing, may (instead of being any other equivalent 
group) be that group which contains the before-mentioned substitution 
a = abc. def. ghi .jkl, and /3 = ajg. bif. cek . dhl. 
Observe that in the diagram, considering the lines to be drawn as shown by the 
arrows, there is from any given point whatever only one black line, and only one red 
line. Let B denote motion along a black line, R motion along a red line (always 
from a point to the next point); then B 2 will denote motion along two black lines 
successively, BR (any such symbol being read always from right to left) will denote 
motion first along a red line, and then along a black line, and so in other cases; a 
symbol or “route” ...R p B a has thus a perfectly definite signification, determining the 
path when the initial point is given. 
The diagram has the property that every route, leading from any one letter to 
itself, leads also from every other letter to itself; or say a route leading from a to 
a, leads also from b to b, from c to c,..., from l to l; and we can thus in the 
diagram speak absolutely (that is, without restriction as to the initial point) of a 
route as leading from a point to itself, or say as being equal to unity; it is in virtue 
of this property that the diagram gives a group. 
For, assuming the property, it at once follows (1) that two routes, each leading 
say from the point a to the same point f lead also from any other point b to 
one and the same point g. Such routes are said to be equivalent, or equal to each 
other; and the number of distinct routes (including the route unity) is thus equal 
to the numbers of the letters, viz. we have only the routes from a to a, to b, ..., to l, 
respectively; (2) a route, leading from a point a to a point f leads from any other point 
b to a different point g; and (3) two routes, leading from the same point a to different 
points b and c, lead also from any other point f to different points k and l. Hence a 
given route leads from the several points abc...I successively to the same series of points 
taken in a different order, or we thus obtain a new arrangement of the points; and 
dealing in this manner successively with the routes from a to a, to b,..., to l, we 
obtain so many distinct arrangements, beginning with the letters a, b, c,..,l respectively, 
such that in no two of them does the same letter occupy the same place; we thus 
obtain a square of 12 such as that already written down, and which is, in fact, the 
same square, the several routes of course corresponding to the substitutions of the 
square. The hemihedron thus gives the foregoing group of 12. 
Observe that the diagram is composed of the four black triangles representing 
the substitution abc. def. ghi. jkl, and of the four red triangles representing the sub 
stitution ajg. bif. cek. dhl; viz. these are independent substitutions which by their powers 
and products serve to express all the substitutions of the group; that they are sufficient 
appears by the diagram itself, in that every point thereof is (by black and red lines) 
connected with every other point thereof. The group might have contained three or 
more independent substitutions, and the diagram would then have contained the like 
number of differently coloured sets of lines. The essential characters are that the lines 
of any given colour shall form polygons of the same number of sides (but for different