Mr. A. Cayley on the Theory of Gi'oups. 409 



say that the holder operated upon by, or operating upon, a sym- 

 bol a of the group reprodiices the holder. 

 Suppose now that the group 



1> «; A Y, 8, e, ?• . 



can be divided into a series of symmetrical holders of the smaller 

 gi-oup 



1, «,/3,.. 



The former group is said to be a multiple of the latter group, 

 and the latter group to be a subraultiple of the former group. 

 Thus considering the two different forms of a group of six, and 

 first the form 



the group of six is a multiple of the group of three, 1, a, a^ (in 

 fact, 1, a, a^ and y, ya, yct^ are each of them a symmetrical 

 holder of the group 1, u, a.^) ; and so in like manner the group 

 of six is a multiple of the group of two, 1, y (in fact, 1, y and 

 a, ay, and «, a^y are each a symmetrical holder of the group 1, y). 

 There would not, in a case such as the one in question, be any 

 harm in speaking of the group of six as the product of the two 

 groups 1, «, «2 and 1, y, but upon the whole it is, I think, 

 better to dispense with the expression. 



Considering, secondly, the other form of a group of six, viz. 



^, <^> «^ y, y*-, ya^ («^=1, y^=l, a.y = yo?). 



Here the group of six is a multiple of the group of three, 1, a, a^ 

 (in fact, as before, 1, a, o? and y, ya., yo? are each a symmetrical 

 holder of the group ] , a, o?, since, as regards y, ya., ya?, we have 

 (y, y«, y(^) =y(l, a, c?) = (1, a^, a)y). But the group of six is 

 not a multiple of any group of two whatever ; in fact, besides 

 the group 1, y itself, there is not any symmetrical holder of this 

 group 1, y ; and so, in like manner, with respect to the other 

 groups of two, 1, ya and 1, ya^. The group of three, 1, a, u^, is 

 therefore, in the present case, the only submultiple of the gi-oup 

 of six. 



It may be remarked, that if there be any number of ^ ymme- 

 trical holders of the sainc group, 1, «,/?,.. then any one of these 

 holders bears to the aggregate of the holders a relation such as 

 the submultiple of a group bears to such group : it is proper to 

 notice that the aggregate of the holders is not of necessity itself 

 a holder. 



