44 Mr. A. Cayley on the Theory of Groups, 



root which is a prime root of ^^=1, or (to use a simpler expres- 

 sion) a root having the index 3. It is clear that if there were a 

 pi'ime rootj or root having the index 6, the square of this root 

 would have the index 3, it is therefore only necessary to show 

 that it is impossible that all the roots should have the index 2. 

 This may be done by means of a theorem which I shall for the 

 present assume, viz. that if among the roots of the symbolic 

 equation ^"=1, there are contained a system of roots of the sym- 

 bolic equation ^''=1 (or, in other words, if among the symbols 

 forming a group of the order there are contained symbols form- 

 ing a group of the order p), then jj is a sub multiple of n. In 

 the particular case in question, a group of the order 4 cannofe 

 form part of the grou]) of the order G. Suppose, then, that 7, S 

 are two roots of ^'= 1, having each of them the index 2 ; then 

 if 7S had also the index 2, we should have 'yB=By; and 1, 7, 

 8, Sy, which is part of the group of the order 6, would be a 

 group of the order 4. It is easy to see that 78 must have the 

 index 3, and that the group is, iu fact, 1, 7S, Sy, y, 8, ySy, which 

 is, in fact, one of the groups to be presently obtained ; I prefer 

 commencing with the assumption of a root having the index 3. 

 Suppose that a is such a root, the group must clearly be of the 

 form 



1, «, «^, 7, ay, a^y, {(X.^=l); 



and multiplying the entire group by 7 as nearer factor, it becomes 

 7, ay, airy, y~, arf-, a^y- ; we must therefore have 7^ = 1, u, or u^. 

 I3ut the supposition y^T^o? gives 7'*=«'' = «, and the group is 

 in this case 1, 7, y^, y^, 7"*, 7^ (7^ = 1) ; and the supposition 

 7^=« gives also this same group. It only remains, therefore, 

 to assume 7^ = 1 ; then we must have either ya.=^a.y or else 

 yu = oi^y. The former assumption leads to the group 



1, «, «-, 7, ay, a^y, (a^rzl, 7^=1, ya — ay), 



which is, in fact, analogous to the system of roots of the ordinary 

 equation .^''— 1=0; and by putting a7=A,, might be exhibited 

 in the form 1, \, \^, \^, X*, \^, [\^ = 1), in which this system 

 has previously been considered. The latter assumption leads to 

 the group 



I> », «■', y, «7; «"% («^ = lj 7"=lj ya. — a^y). 

 And we have thus two, and only two, essentially distinct forms 

 of a group of six. If we represent the first of these two forms, 

 viz. the group 



1; «, «^ 7> «7j «^7j («^= ij 7^= !;■ 7« = «7) 

 by the general symbols 



1. », A % ^, f, 

 we have the table 



