132 ON THE THEORY OF GROUPS, &C. [126 
The system L, M, N,... may be termed a group-holding system, or simply a 
holder; and with reference to the two groups to which it gives rise, may be said 
to hold on the nearer side the group L~ l L, L~ X M, L~ X N,..., and to hold on the 
further side the group LL~ l , LM~ X , LN~ X ,... Suppose that these groups are one and 
the same group 1, a, /3..., the system L, M, W,... is in this case termed a sym 
metrical holder, and in reference to the last-mentioned group is said to hold such 
group symmetrically. It is evident that the symmetrical holder L, M, N, ... may be 
expressed indifferently and at pleasure in either of the two forms L, La, L/3,... and 
L, aL, ¡3L; i.e. we may say that the group is convertible with any symbol L of 
the holder, and that the group operating upon, or operated upon by, a symbol L of 
the holder, produces the holder. We may also say that the holder operated upon by, 
or operating upon, a symbol a of the group reproduces the holder. 
Suppose now that the group 
1, a, ¡3, y, 8, e, £ ... 
can be divided into a series of symmetrical holders of the smaller group 
1, a, & ...; 
the former group is said to be a multiple of the latter group, and the latter group 
to be a submultiple of the former group. Thus considering the two different forms 
of a group of six, and first the form 
1, a, a 2 , 7> 7 a > 7 a2 > ( a3 = T 7 2 =T a 7 = 7 a )> 
the group of six is a multiple of the group of three, 1, a, a 2 (in fact, 1, a, a 2 
and y, ya, 7a 2 are each of them a symmetrical holder of the group 1, a, a 2 ); and 
so in like manner the group of six is a multiple of the group of two, 1, 7 (in fact, 
1, 7 and a, ay, and a, a 2 y are each a symmetrical holder of the group 1, 7). 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, a, a 2 and 1, 7, 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. 
1, a, a 2 , y, 7 a j 7a 2 (a 3 = 1, 7* = 1, ay = 7a 2 ); 
here the group of six is a multiple of the group of three, 1, a, a 2 (in fact, as be 
fore, 1, a, a 2 and 7, ya, ya 2 , are each a symmetrical holder of the group 1, a, a 2 , 
since, as regards 7, ya, ya 2 , we have (7, ya, ya 2 ) = y(l, a, a 9 ) = (1, a 2 , a) 7). But 
the group of six is not a multiple of any group of two whatever; in fact, besides 
the group 1, 7 itself, there is not any symmetrical holder of this group 1, 7; and 
so, in like manner, with respect to the other groups of two, 1, ya, and 1, 7a 2 . The 
group of three, 1, a, a 2 , is therefore, in the present case, the only submultiple of 
the group of six. 
It may be remarked, that if there be any number of symmetrical holders of the 
same group, 1, a, /3, ... 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.