No. 2.) 



TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 



29 



aro shown by lines separating the triads in a class. The system VI347 contains 1 1 nonpermutable 

 subdivisions given in the following table; 



In determining the group we examine first for substitutions that transform into itself one 

 of the trains, and secondly for those that transform this train into the remaining trains of its 

 class. Substitutions when determined must be tested on the 35 triads of the system. 



The substitutions may be determined from any class of trains in the system, but most 

 easily from the class containing the trains with the greatest nimiber of triads since these exhibit 

 more repetition of the elements. In the present case (4), figure 206 is selected. 



/26~. 

 abf — cg8- 



-457n 



alg- 



■ c/7 368- 



ide 



dbe — c<Z6 285s 



d48-^ 



ahd—ceb 176- 



(i) Examine for substitutions to transform the train ode into itself. 

 two similar parts and the substitution 



t={a) (b) (c) ide) ifg) (12) (34) (56) (78) 



-afg 



The train consists of 



permutes these similar parts. No other substitution exists which converts this train ade into 

 itself. Therefore only a subgroup of order 2 transforms this train into itself. 



(ii) Examine for substitutions to transform the train of the triad ade into the other train 

 of its class. The substitution 



s=(a) (6) (c) {dfeg) (1423) (5867) 



transforms the train of ade into the train of afg. Since t^^ the substitution t is omitted. The 

 substitution s applied to the 35 triads of the system transforms the system into itself. There- 

 fore the group for this system is a cycUc group of order 4 and is generated by s^{a) (b) (c) 

 (dfcg) (1423) (5867). 



Similar detailed study determines the group for each of the following 43 systems: 



Trains for the System V4al. 



Ten classes of trains terminating in triads of the system: (1) Seven trains, figure 1; (2) 6 

 trains, figure 2; (3) 3 trains, figure 3; (4) 3 trams, figure 6; (5) 1 train, figure 66; (6) 3 trains, 

 figure 205; (7) 3 trains, figure 207; (8) 3 trains, figure 208; (9) 3 trains, figure 209; (10) 3 trains, 

 figure 212. 



One class of trains terminating in a cycle of period 12: (11) Three trains, figure 216. 



Group for the System V4al. 



The sets of transitive elements are A; B; C; afl^a^; bfifi^; CtC^c^: d^d^d^. These with 

 the trains separate the system into 13 nonpermutable subdivisions. The group is generated 

 by s^A) {B) (C) (aiajfflj) (bjf^is) (C1CJC3) (did^^j), and is of order 3. 



The trains w^hich belong to the system "V13,7 are exhibited in Plato I, those of the system 

 V4ol in Plate II; even a casual inspection of these two plates establishes conclusively the 

 noncongruence of the two systems. 



The same type of train may occur in several systems, and in order to avoid repetition 

 in the diagrams the 204 distinct types which occur in the remaining 42 sj'stems are numbered 

 and listed in a definite order. The most convenient arrangement seemed to be the following, 



