AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 301 



For operators of the second set with a =(= 0, r =(= or 1 , we prove that 



S a TS a TS t T - T = S 0l TS ai TS tl T, 



where o lf a,, T X are suitably chosen marks, c^ =}=0. The equivalent 

 condition 8.18,18^18^18.^.1=1 



may be satisfied by c) by proper choice of r 1? a l9 G ly with 



G^EEar-1 4=0. 



We next apply S Q as a right-hand multiplier. S a TS a TSa-iS<> 

 will be of the form S 0l IS ai IS tl T s , and consequently belong to the 

 sets by the previous proof, if we have 



Since ^(a^ + e) = 1 + KQ 4= 1? ^ s condition is of the form c) if 

 a i> i> r i be suitably chosen. If Q = a/(ar 1), so that at 4= 1> 

 we have, by c), 



O T 1 C 'FC T 1 Q C T'O T'Qf 



/JO JL /!) JL O|;-L ' IOQ = & 1 f -L Uat 1 J- O 1 . 



a +^=i ^zi 



For the case A~ a Q(CCX 1)=|=0, we prove that 



If ar=^=\j we replace TS a TS^T by its equivalent derived from c) 

 and find that condition f) becomes 



at 1 at 1 



and hence is satisfied from c). If, however, ar = l, so that A=a, 

 then f) takes the simpler form 



f ') TS a TS a -iTS Q = S a -2 Q TS a TS a -iT. 



If also Q =(= a, we replace TS a iTS Q by its equivalent derived from c) 

 and find the condition, where v = a" 1 ^ 1 



This reduces to the identity c) for I a~ 2 Q, /u = a, whence 

 In particular, f ') is true if Q = a -f- 5c ? x =(= 0, so that 



The products in the parentheses are identical and so f) is true for 

 Q = a, if the following condition be true for any particular mark jc =(=0, 



The latter is of the form f ' ) for Q = K and hence is true if x =J= a. 

 But marks x =f= 0, a exist if jp w > 2. For p n = 2, a = 1, so that f ') 

 is true for any Q by d). 



