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ON THE THEORY OF GROUPS. 
[690 
colours the polygons may have different numbers of sides; in particular, for any given 
colour or colours, the polygons may be bilaterals, represented each by a line with a 
double arrow pointing opposite ways); that there shall be from each point only one 
line of the same colour; that every point shall be connected with every other point; 
and finally, that every route leading from one point to itself shall lead also from 
every other point to itself. When these conditions are satisfied the foregoing 
investigation in fact shows that the diagram, or say the hemihedron, gives rise to a 
group. 
It may be remarked that we can, if we please, introduce into the diagram a set 
of lines of a new colour to represent any dependent substitution of the group; thus, 
in the example considered, a substitution is aeh. bjd. oil -fkg, and if we draw these 
triangles in green (the arrows being from a to e, e to h, h to a, &c.), then there 
will be from each point one black line, one red line, and one green line; any route 
...G y RPB a will thus be perfectly definite, and will have the same properties as a route 
composed of black and red lines only; and the theory thus subsists without alteration. 
I remark, in conclusion, that the group of 12 considered above is, in fact, the 
group of 12 positive substitutions upon 4 letters abed-, viz. the substitutions are 1, 
abc, acb, abd, adb, acd, adc, bed, bde, ab. cd, ac. bd, ad .be; the groups each contain 
unity, three substitutions of the order (or index) 2, and 8 substitutions of the order 
(or index) 3, and their identity can be easily verified.