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

CHAPTER XX

MONODROMIE GROUP
193. Definition of the Monodromie Group M. Consider
an algebraic equation F(z, k)=0 in z whose coefficients are
rational functions of the complex variable k. Let zi,..., Z
be the roots of F(z, ko) = 0, where ko is a constant. Let k vary
continuously from this initial value in any manner, but finally
return to the same value ko (i.e., let the point representing k
in the complex plane describe any closed path starting from and
ending with the point representing ko). Then the roots vary
continuously and, after the circuit, take on their initial values
in the same or a new order. Thus to each closed path corresponds a substitution on the roots.
For example, if k describes a circle around the origin, the
roots of 2 = 2k are interchanged.
Two circuits may be combined into a single third circuit
to which corresponds the product of the two substitutions
corresponding to the two circuits. Hence the substitutions
corresponding to all possible circuits form a group M, called
the monodromie group * of F(z, k)=0 with respect to k. It
was first studied by Hermite and Jordan.t
194. Monodromie Group an Invariant Subgroup of the
Galois Group. Let ~ be a rational function of k and the roots
Z1,..., zn, and let 0, 4', "/,... be the functions derived
from ( by the various substitutions of M. If ~ = ' = /=..
4 is said to possess monodromie with respect to k. This is
evidently the case with any 4 which equals a rational function
*" Group in the function-theoretic sense," by Klein-Fricke, Elliptischen
Modulfunctionen, vol. 1, 1890, p. 132; applications in vol. 2, 1892, p. 53, p. 599.
t Traite des Substitutions, 1870, pp. 277-9.
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