887] 
639 
887. 
ON THE THEORY OF GROUPS. 
[From the American Journal of Mathematics, vol. XI. (1889), pp. 139—157.] 
I refer to my papers on the theory of groups as depending on the symbolic 
equation 6 n = 1, Phil. Mag., vol. vn. (1854), pp. 40—47 and 408, 409, [125, 126]; also 
vol. xviii. (1859), pp. 34—37, [243]; and “ On the Theory of Groups,” American 
Journal of Mathematics, vol. i. (1878), pp. 50—52, and “The Theory of Groups: 
Graphical Representation,” ibid., pp. 174—176, [694]; also to Mr Kempe’s “Memoir 
on the Theory of Mathematical Form,” Phil. Trans., vol. clxxvii. (1886), pp. 1—70, 
see the section “Groups containing from one to twelve units,” pp. 37—43, with the 
diagrams given therein. Mr Kempe’s paper has recalled my attention to the method 
of graphical representation explained in the second of the two papers of 1878, and 
has led me to consider, in place of a diagram as there given for the independent 
substitutions, a diagram such as those of his paper, for all the substitutions. I call 
this a colourgroup; viz. for the representation of a substitution-group of « substitutions 
upon the same number of letters, or say of the order «, we employ a figure of « 
points (in space or in a plane) connected together by coloured lines, and called a 
colourgroup. 
I remark that up to «=11, the first case of any difficulty is that of « = 8, 
and that the 5 groups of this order were determined in my papers of 1854 and 
1859. For the order 12, Mr Kempe has five groups, but one of these is non-existent, 
and there is a group omitted; the number is thus = 5. 
The colourgroup consists of « points joined in pairs by ^«(« — 1) coloured lines 
under prescribed conditions. A line joining two points is in general regarded as a 
vector drawn from one to the other of the two points; the currency is shown by 
an arrow, and in speaking of a line ab we mean the line from a to b. But we 
may have a line regarded as a double line, drawn from each to the other of the 
two points; the arrow is then omitted, and in speaking of such a line ab we mean 
the line from b to a and from a to b. A fresh condition is that for a given 
colour there shall be one and only one line from each of the points, and one and