ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 65 



Suppose that this condition is satisfied by two functions M^(X) and 

 and consider the effect of applying first the substitution 1 ) 



A: X'=M 



and afterwards the substitution 



TO 

 -p. -\rfl \lt /"Vl\ "^1 T> -y-f w 7l(TO ,;') 



Jj. -A- = T^^^L J > AwJL p 



^J 



The result is equivalent to that produced by the single substitution 



1, ...,TO 



-.7') V* (2 m/;) 



After reduction by means of X* nm =X, this equation may be written 



(7* X" = Y (X) = 



= 1 



each (7; being a definite function of the A/s and B/s. By hypothesis, 



requires ^(X) = 0, which in turn requires X = 0. Hence, X= 

 is the only solution in the field of H'c(X) = 0. It follows that the 

 transformation C represents a substitution upon the marks of the 

 GF[p n ]. C is called the compound, or product, of A and B, and 

 the above relation is expressed in the symbolic form, 



C = AB. 



Giving to the coefficients At every possible combination of values 

 in the GF[p nm ~\ such that 



TO 



~V"f "^^T A -f7-j,n(m i) 

 ^J 



represents a substitution on its marks , we obtain a set of substitutions 

 having the property that the result of applying first any one of the 

 set and afterwards any one of the set is identical with the result 

 of applying a single substitution of the set, called the product of 

 the two. Such a set of substitutions is said to form a group. In 

 the present case, the group will be called the Betti-Mathieu Group. 2 ) 



1) The present notation is used in place of and as equivalent to 



2) For n = 1, this group was studied by Betti, Annali di Scienze Mat. e 

 Fisiche, vol. 3 (1852), pp. 49115, vol. 6 (1855), pp. 6 34; for general n, by 

 Mathieu, Jownal de Math., (2) vol. 5 (1860), pp. 9 42, vol. 6, pp. 241323. 

 The theorems of 92 94 are due to the author, Annals of Math., (1897), 

 pp. 94 96, 178-183. 



DlCKSON, Linear Groups. 6 



