GROUP INVAUIANCE OU TOTAL SYMMETRY 



1449 



The matrix can therefore be further partitioned by separating the rows 

 into groups of rows which have the same weight in every cokmin parti- 

 lion, Table 11(c). Similar remarks hold for the columns so that it may 

 then be necessary to partition the columns again so that each column in 

 a partition has the same weight in each submatrix, Table 11 (d). Par- 

 titioning the columns may make it necessary to again partition the 

 rows, which in turn may make more column partitioning necessary. This 

 process should l)e carried out until a matrix results in which each row 

 (column) of each submatrix has the same weight. Inspection is then 

 used to determine which row and column permutations will leave the 

 matrix unchangetl. Only permutations among rows or columns in the 

 same partition need be considered. 



From the matrix of Table 11(d) it can be seen that permuting either 

 columns .r^ and .r4 or columns x^ and x^' will not change the matrix aside 

 from reordering certain rows. This means that interchanging .T3 and X4 

 or priming and interchanging X5 and x^ in the original transmission will 

 leave the transmission unchanged. Interchanging x^ and .T5 means re- 

 placing X5 by xt and x^ by x^,' which is the same as interchanging x^ and 

 x% and then priming both Xi, and Xq . Thus for the transmission of Table 

 II 0124356-Z = T and A* 000011*^123465-^ = N^Sus^ebT = T. 



A procedure has been presented for determining the group invariance 

 of any transmission matrix which does not have an equal number of I's 

 and O's in any column. This must now be extended to matrices which do 

 have equal numbers of O's and I's in some columns, Table Ill(a). For 

 such matrices the procedure is to prime appropriate columns so that 

 there are either more O's than I's or the same number of O's and I's in 

 each column, Tal)le Ill(a). This matrix is then partitioned as described 

 above and the permutations which leave the matrix unchanged are de- 

 termined. The matrix of Table Ill(a) is so partitioned. Interchanging 



