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

~ 150]

TRANSITIVE GROUP

289

G(Vl), being equal to the number zero in R, remains unaltered
in value by the substitutions a, 3,..., x, so that
0=G(V1)=G(Va)=... =G(Vx)
Hence V,..., Vx occur among the roots (6) of G(V)=0.
Thus r is a subgroup of G.
If G'= r, then G'=G.
In view of its repeated application below, we state our
second result as the
COROLLARY. If every rational function of the roots with
coefficients in R which equals a quantity in R is unaltered in
value by every substitution of a group r, then r is a subgroup of
the group G for R of the equation.
150. Transitive Group. We shall make much use of the
THEOREM. If an equation is irreducible in a domain R,
its group for R is transitive, and conversely.
Consider an equation f(x) = 0 irreducible in R. Contrary
to the theorem, suppose that its group G for R is intransitive
and contains substitutions replacing xi by xi, x2,..., m,
but none replacing xi by one of Xm+,., x.. Hence
every substitution of G permutes xl,..., xm amongst themselves and thus leaves unaltered any symmetric function of
them. Hence
g(x)-(x-Xi)(x-X2)... (x-Xm)
has its coefficients in R, in view of property A. Thus f(x)
has the factor g(x) in R, contrary to its irreducibility in R.
To prove the converse, let G be transitive and the equation
f(x) = 0 be reducible in R. Let the preceding function g(x)
be a factor of f(x), the coefficients of g(x) being in R and its
degree m being less than n. Since g(xl) equals the number
zero of R, it is unaltered by every substitution of G (property
B). Since G is transitive, g(xi)=0 for i=1,..., n. This
contradicts m< n.
EXAMPLE 1. Find the group G of x3-7x+7=0 for R(1).
The equation is irreducible (Ex. 4, ~ 144), so that G is transitive. It