SyNTIIKSLS O/' SWITCHING CIKCmi'S 93 



forming on i, considered as an ordered sequence of zero's and one's, the 

 permutation Sj . Thus 



liy tliis rule an\' product such as N' i Sj N k N i Sm ^^ n S p can be reduced to the 

 form 



A^•iVy ••• N„SpS, ■■■ Sr 



and this can then be reduced to the standard form A^iSj . 



A function/ will be said to have a non-trivial group invariance if there are 

 elements XiSj of G other than / such that identically in all variables 



N,S, f = f. 



It is evident that the set of all such elements, NiSj , for a given function, 

 forms a subgroup Gi of G, since the product of two such elements is an ele- 

 ment, the inverse of such an element is an element, and all functions are 

 invariant under /. 



A group operator leaving a function / invariant implies certain equalities 

 among the terms appearing in the expanded form of /. To show this, 

 consider a fixed I^^iSj , which changes in some way the variables (say) 

 Xi , X2 , • • • , Xr . Let the function f{Xi , • • • , X„) be expanded 

 about Xi , ■ ■ • , Xr : 



/ = [Xi + X2 + • • • + A% + /i(Z,+i , • • • , X„)] 



[X[ + X2+ ■■■ + Xr + MXr+l ,■•■ , Xn)] 

 [X[ + X: + • • • + Xl + MiXr+l ,■■• , Xn)] 



If/ satisfies XiSjf — f we will show that there are at least j2'' equalities 

 between the functions fi,fo, ■ • • , f-ir. Thus the number of functions 

 satisfying this relation is 



since each independent /, can be any of just 2~ functions, and there are 

 at most f 2'' independent ones. Suppose A^^*; changes 



Xi , X2 , • • ' , Xr A 



into 



Xai , Xaj , • • • , Xar B 



where the *'s may be either primes or non primes, but no Xa, = A', . Give 



