PART 1. 



TRIAD SYSTEMS ON 15 ELEMENTS WHOSE GROUP IS OF ORDER HIGHER 



THAN UNITY. 



By H. S. White. 



§ 1. INTRODUCTION. 



The problem which first led to the study of triad systems (Tripel-systeme) was proposed 

 in the first place for 15 elements, Kirlonan's "Fifteen schoolgirls problem." ' In various 

 journals, during the past 60 j^ears, are described many different ways of consti-ucting triad 

 systems on 15 things. There was not known, prior to 1912, any short and decisive method 

 for comparing systems apparently different; accordingly duplicates were produced, and up to 

 1912 only 10 noncongruent systems had been found. A triad system, hke any other airange- 

 ment of elements, may have its appearance changed while its structure is unaltered by a per- 

 mutation among the elements — a substitution. When one system can be derived from another 

 by a substitution, the two are called congruent or equivalent; otherwise they are incongruent or 

 nonequivalent. If there are substitutions which transform a system into itself (usually per- 

 muting its triads among themselves), aU such substitutions together are called the group of 

 that triad system. 



The number of elements in a triad system must be of the form 6n+l or Gn + 3, where n 

 is an integer.^ Such numbers are 3, 7, 9, 13, 15, 19, etc. For any number of elements under 

 15 the exact number of nonequivalent triad systems has been known for some time, namely, 

 one S3^stem for each of the numbers 3, 7, 9, while for 13 there are two systems. These two 

 systems on 13 elements have different groups, of orders 6 and 39, respectively. One might 

 anticipate that for numbers above 13 the same thing might happen, so that the group would 

 serve as a distinctive mark or characteristic for its sj^stem. Miss Cummings has shown, hovr- 

 ever, that for 15 the case is different; that sometunes the same group belongs to two or more 

 Incongruent S3^stems.' 



.(Vnother test has been employed by E. H. Moore to prove the nonequivalence of two 

 systems.^ K among the 35 triads in 15 elements there occur 7 triads on any 7 elements, exclu- 

 sively, the larger system, which Moore denotes by A15, is said to contain the smaller, a A,. 

 Then if one Au does contain a A7, while another does not contain any A7, the two are obviously 

 incongruent. But this is of course not a conclusive test for equivalence, since Miss Cummings 

 has found 23 incongruent Aij's, each of which does contain a A,. 



Probably it has been the lack of convenient and rehable tests for equivalence or non- 

 equivalence that has deterred investigators from the task of finding how many essentially 

 distinct triad sj^stems are possible in 15 elements. But now we have available two distinct 

 methods of comparison, both of which have given rehable results, positive as well as negative, 

 in all cases where they have been tried, although then- value as positive tests of equivalence 

 still lacks a priori demonstration. One of these, the method of trains, uses a given triad system 



' T. p. ICirkman: On a problem in combinations. Cambridge and Dublin Mathematical Journal, vol. 2 (1847), p. 191. See also Note on an 

 unanswered prize question, ibidem, vol. 5 (1850), p. 255. 



• Nctto: Substautiotunlheorie, p. 220, 5 192. 



• L. D. Cummings: Note on the omupafot triple systems. Bulletin of the American Mathematical Society, vol. 19 (1913), p. 355. 



• E. Hastings Mooro; Conurning triple at/stems. Mathematische Annalen, vol. 43 (IS93), p. 271. 



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