36 Theojy of Groups depending on Symbolic Equation 6" =1. 

 which is equal to 



or finally to 



Qb + d■^fg^as''+f+csf+e^ 



And the result would have been precisely the same if, instead of 

 thus combining together the first and second factors and the 

 product with the third factor, we had combined the first factor 

 with the product of the second and third factors, so that the 

 associative law is satisfied. 



It is now easy to see that if, as before, 



«'«=!, /3" = 1, ci/3=/3u% 

 conditions which it has been shown imply s" = l (mod. m), then 

 the symbols /8V (or, if weplease, a''/S'), where p has the values 

 0, 1, 2 . . . (m— 1), and q the values 0, 1, 2 . . .n — 1, form a 

 group of mn terms. In particular, as already noticed, if n = 2 

 and m is prime, then s = l or s = m — l ; the two groups so ob- 

 tained are essentially distinct from each other. If ?i = 2, but m 

 is not prime, then s has in general more than two values : thus 

 for ?w=12, s^=£l (mod. 12), which is satisfied by s = l, 5, 7, 

 and 11 ; the group corresponding to s=l is distinct from that 

 for any other value of s, but I have not ascertained whether the 

 values other than unity do, or do not, give groups distinct from 

 each other. 



For the sake of an observation to which it gives rise, I write 

 down an example of a group corresponding to ra=2, s=m — l, 

 say m = 5, and therefore s=s4, so that we have 



And the group is 



1, «, «^ a3, u^ 13, /3«, /3«^ /3«3, pu\ 

 the indices of the several terms being 



1, 5, 5, 5, 5, 2, 2, 2, 2, 2. 



The group is here expressed by means of the symbols a, /8, 

 having the indices 5 and 2 respectively, but it may be expressed 

 by means of two symbols having each of them the index 2. 

 Thus putting /S« = 7, we find /S^ = l, 7^ = 1, (y8y)^=l, which is 

 equivalent to (7/S)*=l, and the group may be represented in 

 the form 



1, A 7, /37, 7/3, /8y^, 7^7, /S7^7, yfi^^, ^yBy^=y^y^y, 

 the equality of the last two symbols being an obvious consequence 

 of the equation (/S7)^=l. It is clear that for any even number 

 2p whatever, there is always a group which can be expressed in 

 this form. 



