PART 2. 



TRAINS FOR TRIAD SYSTEMS ON 15 ELEMENTS WHOSE GROUP IS OF 



ORDER HIGHER THAN UNITY. 



Bj' L. D. CUMMINGS. 



To investigate the 71 systems obtained in Part 1, and to determine the group for each 

 system, Mr. White's method of comparison ' for triad systems is employed. For this method 

 the triple system is regarded as an operator and certain covariants of that operator are deduced. 

 Those covariants can bo represented graphically and are called the trains of the system. 



The trains show that the 71 systems are reducible to 44 noncongruent systems; of these 

 24 are completely known systems already fully discussed in my dissertation,^ but the remaining 

 20 systems have not been described heretofore. The trains for the 44 noncongruent systems 

 are exhibited, and for each of the 20 new systems the group is determined. The substitutions 

 which transform the 51 systems into their equivalent systems are also given below. 



A triple system on n elements consists of triads so selected that every pair of elements 

 (or dyad) occurs once and only once in the chosen triad. If there are 15 elements, every 

 element occurs with 7 pairs of others, and there are in the system in all 35 triads. This property 

 qualifies the triad system to be a transformer of dyads into single elements, and since each 

 dj'ad occurs once and no more this duality is unique for dyads. Thus, if the system contains 

 the three triads 124, 135, 236, then it will transform the triad 123 which contains the pairs 

 12, 13, 23 into the triad 456. 



From 15 elements 455 triads can be formed. Any system contains 35 of these, leaving 

 420 that may be called extraneous triads. Apply the system to transform them all; we shall 

 see, as in the example worked out below, that the 35 triads in the system are transformed into 

 themselves, but the 420 extraneous triads go either into extraneous triads or possibly into 

 triads of the system. Some triads will transform into themselves, some wiU be produced more 

 than once, and others may not be produced at all by the transformation. All that arefomid to 

 be produced by the transformation are called derivative; all that are missing after the transfor- 

 mation, if any, are called primitive. • 



TRAINS OF TRIADS. 



Under a given triad system as an operator, let a triad /, be converted into the triad t^. 

 Repeat the operation and continue indefinitely, so that /j becomes t^; t^ becomes t^. Since only 

 455 triads exist, either a triad of the system t^ is reached or else a triad that has already appeared 

 is repeated, namely, tm+k^tm- In the former case the triad tj. repeats forever, while in the 

 latter case the train beginning at tm constitutes a recurring c}'cle. If the triads of the system 

 are designated as one-term cycles, then every triad that is primitive with respect to a given 

 triple system initiates a train terminating in a periodic cycle. Triads that do not recur in the 

 terminal cycle are classified as forming appendices, and a complete train consists of one recurrent 

 cycle and aU its appendices. 



Some substitution may transform the triple system into itself. Such a substitution evi- 

 dently must also transform each train into itseK or into a precisely similar train and therefore 



' H. S. White: Triple systems as transronnations and their paths among triads. Transactions of the American Mathematical Society, vol. 14 

 (1913). pp. 6-13. 



' I.. D. Cummings: On a method of comparison for triple systems. Transactions of the American Mathematical Society, vol. IS (1914), pp. 

 311-327. 



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