'>2 BELL :iYSTEM TECUNICAL JOURNAL 



in all variables, has been considered in (6). A function /(Xi , ••• , Xn) 

 will be said to be symmetric in .Vi , .Y2 if it satisfies the relation 



f{\\ ,X,,---, A'„) = f{X, . A'l , • ■ • , Xn). 



It is symmetric in A'l and A''2 if it satisfies the equation 



fix, ,X,,---, Xn) = f{X', ,X[,X,,--- ,Xn) 



These also are special cases of the type of functional relationships we will 



consider. Let us denote by 



Xoo ••■0^1 the operation of leaving the variables in a function as they 



are, 

 Xioo • • • o the operation of negating the first variable (i.e. the one occupy- 

 ing the first position), 

 A^io • • • o that of negating the second variable, 

 Xuo ■ ■ • o that of negating the first two, etc. 

 So that NioifiX, V, Z) = f(X'YZ') etc. 



The symbols Ni form an abelian group, with the important property that 

 each element is its own inverse; NiNi = / The product of two elements 

 may be easily found — if A^- Nj = Nk , k is the number found by adding i 

 and j as though they were numbers in the base two but wilhotd carrying. 



Note that there are 2" elements to this "negating" group. Now let 

 ■5'i,2,3,...,w = I = the operation of leaving the variables of a function in the 



same order 

 S2,i,i....n = be that of interchanging the first two variables 



'S'3,2,i,4,....w = that of inverting the order of the first three, etc. 



Thus 



SnoJiX, Y, Z) = fiZ, X, Y) 



SsufiZ, A, ]') = Sl,f(X, Y, Z) = f{Y, Z, X) 



etc. The Si also form a group, the famous "substitution" or "symmetric" 

 group. It is of order n !. It does not, however, have the simple properties 

 of the negating group — it is not abelean (w > 2) nor does it have the self 

 inverse property.* The negating group is not cyclic if n > 2, the symmetric 

 group is not if n > 3. 



The outer product of these two groups forms a group G whose general 

 element is of the form A\5> and since i may assume 2" values andj, n I values, 

 the order of G is 2"«1 



It is easily seen that SjNi = NkSj, where k may be obtained by per- 



* This is redundant; the self inverse property impUes commutativity for if A'A' = / 

 thenXF = (XF)-' = F-^X"' = YX. 



