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

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THEORY OF EQUATIONS

110. Normal Sub-groups of Prime Index. Of special interest
in the theory of equations are the series of groups
PIy P2,,... Pi) Pi+l, ***, 1
so related to each other that each group Pi,, is a normal subgroup of the preceding group Pi the index of Pi+, under P,
being a prime number. Such an assemblage of groups is called
a principal series of composition. If the restriction of a prime
index is removed, then the assemblage is called simply a series
of composition.
Ex. 1. Show that a principal series of composition is (a) for groups of
the third degree, G6(3), 6(3), 1, (b) for groups of the fourth degree, G4(4),
G12(4), G4(4) II, G2(4), 1.
Ex. 2. Show that, for the group of the fifth degree G0(5), a principal
series of composition is G20(5), Go0(5), G5(5), 1.
Ex. 3. Show that G^(4) II is a normal sub-group of Gs(4), G12(4), and
G24(4).
111. Functions which belong to a Group. When G1 is a subgroup of G, a rational function of n letters a, a2, ***, a,( is said
to belong to G1, if the function is unaltered in value by the substitutions of G,, but is altered by all other substitutions of G.*
* If the coefficients of f () = 0 are independent variables, then its roots
are independent of each other. A function of the roots must therefore be
looked upon as having an alteration in value whenever the function experiences an alteration in/o/rm. In other words, when the roots are independent
of each other, two functions of these roots are equal to each other only when
they are identically equal. In the present chapter the roots are so taken.
When the coefficients of f(x)= 0 represent particular numerical values,
its roots are fixed values. Two functions of these roots may be numerically
equal to each other even when they have different forms. Hence, in an equation
whose coefficients have special values, a function of the roots may be formally
altered by a substitution and yet experience no change in numerical value.
Take, for instance, the equation with special coefficients, x3 = 1. If w is one
of its complex roots, we may write 0o ==, a1 == 2, a2 = O3. The function
(X02(1 is altered in form by the substitution ((t0a2(1), but not in value; for,
o02atl = (t2o = o. That functions of ct0, (t, ct2, may have different forms,
but the same numerical value is seen also in the equalities
ao02a2 = aGla X= a = a-02.