144G THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



alent to NwoiT. The permutation and priming operators can be combined. 

 For example, 



S2mN3T(xi , X2 , xs , Xi) = T{x2 , Xi , x^, Xi) 



The symbols SiNj form a mathematical group, ^ hence the term group 

 invariance. 



The problem considered here is that of determining which A^,- and Sj 

 satisfy the relation NiS/F = T for a given transmission T. Since there 

 are only a finite number of different Ni and Sj operators it is possible in 

 principle to compute NiSjT for all possible NiSj and then select those 

 NiSj for which NiSjJ' = T. If T is a function of n variables, there are 

 n! possible Sj operators and 2" .V, operators so that there are n!2" pos- 

 sible combinations of N'iSj . Actually, if NiSjT = T then NiT must 

 equal SjT''^^ so that it is only necessary to compute all NiT and all Sj7\ 

 For /I = 4, n! = 24 and 2" = 16 so that the number of possibilities to 

 be considered is quite large even for functions of only four variables. It 

 is possible to avoid enumerating all NiT and SjT by taking into account 

 certain characteristics of the transmission being considered. 



The first step in determining the group invariances of a transmission 

 is the same as that foi finding the prime implicants.* The binary equiva- 

 lents of the decimal numbers which specify the transmission are listed 

 as in Table 1(a). This list of binary numbers will be called the transmis- 

 sion matrix. When two variables are interchanged, the corresponding 

 columns of the transmission matrix are also interchanged, Table 1(b). 

 When a variable is primed, the entries in the corresponding column of 

 the transmission matrix are also primed, replaced by 1 and 1 replaced 

 by 0, Table 1(c). 



If an NiSj operation leaves a transmission unchanged then the cor- 



* Minimization of Boolean Functions, see page 1417 of this issue. 



