PART 3. 



GROUPLESS TRIAD SYSTEMS ON 15 ELEMENTS. 



By H. S. White and L. D. Cummings. 



All noncongruent systems, A15, with a group having been determined in Part 1, there 

 arises next the qxicstion concerning the possible existence of triad systems on 15 elements with 

 the group identity. Systems whose group is identity, or gi'oupless systems, do not exist for 

 7, 9, or 13 elements. In a paper' ah-eady pubhshed Mr. White has proved the existence of 

 many groupless systems on 31 elements. An investigation given below in some detail has led 

 to the discovery of a considerable number of noncongruent systems on 15 elements with the 

 group identity. Every groupless system on 31 elements whose existence has thus far been 

 demonstrated contains one or more systems Ais and, therefore, is a headed system. On the 

 contrary, every groupless system on 15 elements Ls headless. 



In any triad system the jiaire of elements are more or less interconnected or interlaced. 

 These interfacings may bo determined by applying to the system under consideration a modi- 

 fied form of the method - of examination by sequences and indices. The A 15 is exhibited in a 

 1 5 by 7 array. Each element heads one column ; below it are placed the seven dyads, which, with 

 the element at the head, constitute the triads of the system. Heretofore sequences and indices 

 have been derived from the three columns of a triad in any A15; the same process is now appHed 

 to every pair of columns and yields what may be called the two-column or contracted indices 

 for the system. Since the number of combinations of 15 columns, two at a time, is 105, this 

 number of pairs of columns must be examined unless the group for the system is known and 

 is different from identity. If the group contains an operator of order 7n, then in general m pairs 

 of colunms are examined simidtaneously. The process may be illustrated in its apphcation to 

 a system VII3/3, with a group of order 4 generated by <=(a) (be) (dSel) (jSgi) (1536). Pairs 

 of columns selected from the follomng table show every type of two-column or contracted index 

 that can occur in any system. 



The substitution t applied to the pair of colimins ah gives the pair ac with the same index 

 and similar sequences. If t is apphed to the pair of columns 61, the three pairs c5, l>2, c6 are 

 obtained with index 3". The analysis of the 105 pairs of columns shows that the contracted 

 indices 2'; 2, 4; 3-; 6 belong, respectively, to 2, 24, 4, and 75 pairs of columns. 



1 White, H. S.: Transactions of the American Mathematical Society, vol. 14 (1913), pp. 13-19. 



' Cumtnings, 1,. D.: Transactions of the American Mathematical Society, vol. 15 (1914), pp. 311-327. 



