A CLASS OF BINARY SINGALING ALPHABETS 



227 



It has been remarked that every (n, /v)-alphabet is isomorphic with 

 Ck . Let us suppose the elements of Ci, hsted in a column starting with / 

 and proceeding in order /, 1, 2, 3, • • • , /.', 12, 13, ■••,(/.•— 1)/,-, 123, 



, 123 • • • k. The elements of a given (n, A-)-alphabet can be 



paired off with these abstract elements so as to preserve group multipli- 

 cation. This can be done in many different ways. The result is a matrix 

 with elements zero and one with 7i columns and 2 rows, these latter 

 being labelled by the symbols /, 1,2, • • • etc. What can be said about 

 the columns of this matrix? How many different columns are possible 

 when all (n, A)-alphabets and all methods of establishing isomorphism 

 with Ck are considered? 



In a given column, once the entries in rows 1,2, • • • , /,• are known, the 

 entire column is determined by the group property. There are therefore 

 only 2 possible different columns for such a matrix. A table showing 

 these 2 possible columns of zeros and ones will be called a modular repre- 

 senfafion table for Ck ■ An example of such a table is shown for /,• = 4 in 

 Table VI. 



It is clear that the colunuis of a modular representation table can also 

 be labelled by the elements of Ck , and that group multiplication of these 

 column labels is isomorphic with mod 2 addition of the columns. The 

 table is a symmetric matrix. The element with row label A and column 

 label B is one if the symbols A and B have an odd number of different 

 numerals in common and is zero otherwise. 



Every (n, /c)-alphabet can be made from a modular representation 

 table by choosing w columns of the table (with possible repetitions) at 

 least k of which form an independent set. 



Table VI — Modular Representation Table for Group C4 



I 12 3 4 12 13 14 23 24 34 123 124 134 234 1234 



I 



1 



2 



3 



4 



12 



13 



14 



23 



24 



34 



123 



124 



134 



234 



1234 



