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

380

MONODROMIE GROUP

[CH. XX

the periods of hyperelliptic functions of four periods, the group
is the quaternary abelian linear group modulo 3 and is isomorphic * with the group for the equation of the 27 lines on a
cubic surface (~ 183).
Monodromie has been applied to linear differential equations
dz dn-1z
kn+fl(k) + dk + fn(k) z=0.
For simplicity, the coefficientsf(k) will be assumed to be rational
functions of the complex variable k. Let zi,..., zn be a set of
linearly independent solutions (integrals) and let ko be a constant
such that each zi is an ordinary power series in k - ko. If now the
point representing k describes a closed path starting from and
ending with the point representing ko, as in ~ 193, the set of
solutions Zi,..., zn becomes a set of solutions z',...,
which are therefore linear functions of zl,..., z with constant coefficients:
Ztl =allZl+.. ~ +aln,, z. z nln + ~ ~ +an.nnWith the chosen circuit is thus associated a linear transformation
(~ 75). The transformations obtained from all such circuits
form a linear group, called the monodromie group M of the
differential equation.
This group M is finite in case the integrals Zi,..., Zn are
algebraic functions of k. The theory of finite linear groups
(Part II) is therefore applicable to the problem of the determination of all linear differential equations whose integrals are
all algebraic. In the case just mentioned, M coincides with
another important group G, which we proceed to define; but,
in general, M is only a subgroup of G.
According to Picard and Vessiot, the transformation group
G of a linear differential equation with the linearly independent
solutions zl,..., Zn possesses the following two characteristic
properties (analogous to properties A and B of the Galois group
of an algebraic equation, ~ 149):
If a rational function F of zi,..., zn and their derivatives
* Jordan, I.c., p. 369; Dickson, Linear Groups, pp. 306-7.
t Jordan, Jour.fiir Math., vol. 84 (1877), pp. 89-215.