An introduction to the modern theory of equations, by Florian Cajori.

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THEORY OF EQUATIONS

The powers of the cyclic substitution of 1n letters (ala2 '.' an)
constitute the cyclic group of the degree n.
A function of n letters which is unchanged in value by all
the substitutions of the cyclic group, buzt by no others, is called a
cyclic function. The simplest cyclic function belonging to the
cyclic group of the degree n is
ala22 + a32 + +.. + a,,_lan2 + a,,al2.
Ex. 1. Show that the function aa22 2+ a2a32 + a3a12 belongs to the
cyclic group 1, (aia2a3), (ala3as).
Ex. 2. Show that (ai + a2w + aaw2)3 belongs to the cyclic group of
degree 3, w being a complex cube root of unity.
Ex. 3. By raising (ala2a,3a4) to powers find the cyclic group of the
degree 4.
102. Transitive and Intransitive Groups. In the group
1, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)
the second substitution replaces 1 by 2, the third replaces 1 by
3, the fourth replaces 1 by 4. Similarly, by means of these
substitutions the digits 2, 3, or 4 can be changed into every
other digit operated on by the substitutions in the group. This
group is said to be transitive.
A substitution group is called transitive when it permits any
element to be replaced by every other.
A group that is not transitive is called intransitive. As an
example of the latter we give the following group,
1, (1 3), (2 4), (1 3)(2 4).
Here neither I nor 3 can ever be replaced by either 2 or 4.
103. Primitive and Imprimitive Groups. If in the transitive
group consisting of the six substitutions
1, (1 2 3 4 5 6), (1 3 5)(2 4 6), (1 4)(2-5)(3 6), (1 5 3)(2 6 4),
(1 6 5 4 3 2)