ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 71 



to the case w>l; for SQ[6, 3 2 ] it is necessary to consider, besides 

 the 2 d and 4 th powers, also the power 5 >Y(3 2 l). 



Ex. 3. By the theorem given by Weber, Algebra, II, p. 299, every 

 substitution on p n letters, each affected with n indices #1, #2, . . ., A taken 

 modulo PJ may be represented by the transformation (mod p), 



*! = <t>,-(*i, *2, ., *) (i = 1, 2, . . ., n) 



where each <t> t is a rational integral function with integral coefficients. 

 Apply the method of 84 and show that, on raising each <t> t - to the 

 powers 1, 2, . . ., p 2 and reducing by means of *?= 2,- (mod _p), the 

 coefficient of #? *.* ~ in each power must be congruent to zero. 

 Ex. 4. The following substitution in the G^O 2 "] 



a) X'-A l X? n +AtX (A4=0, .Af+i-^f+H-O) 

 can be reduced to the form Y f = BY by introducing a new index 



b) 7=BiXP+BtX (Bf+iBf+^ti) 



if and only if there exists no root in the G-F[p n ~] of the equation 



^ R A 1 



A? 



= 0. 



If A% + A? n =%= 0, it is not possible to reduce a) to the form 

 Y f = KY P n (K in the aF[p* n ]) 



by a transformation of indices of the form b). 



[The first result is in marked contrast to that of 214 for m = 2]. 



Ex. 5. By the method of 95, show that the sub-groups of the 

 Betti-Mathieu Group defined in 93 94 by means of the invariants Z and Y 

 are identical with certain linear homogeneous groups on m indices in 

 the, G-F[p n ] defined by a linear and a quadratic invariant respectively. 



Ex. 6. (Moore.) The multiplier - GrF [p k ] of the additive - group 

 [Ai, . . ., Aj is the (largest) additive - group common to the additive - groups 



[V*Afc , A" 1 VI (i = 1, . . ., m) 

 and is contained in the p m 1 additive -groups 



[A- 1 A, . . ., A- 1 A m ] (A =f of [A 1? . . ., A m ]). 



