124 
ON THE THEORY OF GROUPS, 
[125 
a supposition which is altogether excluded from the present paper. The order of the 
factors of a product 6$% ... must of course be attended to, since even in the case 
of a product of two factors the order is material; it is very convenient to speak of 
the symbols 0, (f) ... as the first or furthest, second, third, &c., and last or nearest 
factor. What precedes may be almost entirely summed up in the remark, that the 
distributive law has no application to the symbols 6<p ... ; and that these symbols are 
not in general convertible, but are associative. It is easy to see that 6° — 1, and 
that the index law 6 m .6 n =6 m+n , holds for all positive or negative integer values, 
not excluding 0. It should be noticed also, that if 6 = <£, then, whatever the symbols 
a, /3 may be, ad/3 = ac£/3, and conversely. 
A set of symbols, 
1, a, /3, ... 
all of them different, and such that the product of any two of them (no matter in 
what order), or the product of any one of them into itself, belongs to the set, is 
said to be a group\ It follows that if the entire group is multiplied by any one 
of the symbols, either as further or nearer factor, the effect is simply to reproduce 
the group; or what is the same thing, that if the symbols of the group are multi 
plied together so as to form a table, thus: 
Further factors 
1 a /3 
1 
a 
/3 
a 
a 2 
/3a 
& 
a/3 
/3 2 
that as well each line as each column of the square will contain all the symbols 
1, a, /3, .... It also follows that the product of any number of the symbols, with or 
without repetitions, and in any order whatever, is a symbol of the group. Suppose 
that the group 
1, a, /3, ... 
contains n symbols, it may be shown that each of these symbols satisfies the equation 
6 n = l; 
so that a group may be considered as representing a system of roots of this symbolic 
binomial equation. It is, moreover, easy to show that if any symbol a of the group 
1 The idea of a group as applied to permutations or substitutions is due to Galois, and the introduction 
of it may be considered as marking an epoch in the progress of the theory of algebraical equations.