Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

2 EXAMPLES OF GROUPS; DEFINITIONS

[CH. I

This notation implies that each letter is to be replaced by the
one just below it in the same symbol. The same substitutions,
in order, are commonly represented by the following briefer
symbols: *
1, abc, acb, ab, be, ac.
This notation implies that each letter is to be replaced by the
one which follows it in the same symbol, the last being replaced
by the first. Letters which are not replaced are omitted in
this notation, and the symbol for unity is used to represent
the identity; that is, the substitution in which every letter is
replaced by itself.
It is easy to verify the fact that any two of these substitutions, when performed successively, are equivalent to a single
one of them. For instance, if we first apply ab and then ac
the result is the same as if we had applied abc only once. The
process of combining (composing) two substitutions into one is
called multiplication, and it is denoted by the common symbols
for multiplication. Hence abc is said to be the product of ab
and ac. Since ac ab=acb, and ab ac=abc, it results that the
commutative law of multiplication is not always satisfied as
regards the multiplication of substitutions.
A set of distinct substitutions, which has the property that
no additional substitution can be obtained by multiplying
successively each substitution of the set into all the substitutions of the set, is called a substitution group. t Hence the given
set of six substitutions constitutes a substitution group. The
number of the distinct substitutions of a group is called the
order of the group and the number of the distinct letters in its
substitutions is the degree of the group. The totality of the
possible n! substitutions on n letters evidently constitutes a
* These symbols have been called the normal forms of substitutions, J. de
Seguier, Groupes de Substitutions. 1912, p. 3. They are often inclosed in
parentheses.
t The term group in this technical sense is due to E. Galois (1811-32). The
statement that the term group was not used before 1870 with its present technical
meaning, which is found in the Encyclopaudia Britannica, eleventh edition,
vol. 22, p. 626, is incorrect.