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

306

EQUATIONS SOLVABLE BY RADICALS

[CH. XV

spond to the same substitution (3) of r. Hence, by (14) in ~ 154,
we obtain the v distinct substitutions of r by taking those which
correspond to s= 1, g2,..., g. We thus have
THEOREM 3. If H is an invariant subgroup of G of index
v, the group r is a transitive group of order v on v letters and hence
is regular.
COROLLARY. If H is an invariant subgroup of G of prime
index v, then r is a regular cyclic group of order v.
This is illustrated by the above example.
Beginning with the group G of the given equation for the
given domain R, we can find a series of groups G, H, K,...,
G1, terminating with the identity group G1 and such that each
is a maximal invariant subgroup of the preceding. If v is
the index of H under G, p the index of K under H, etc., the factors of composition of G are v, p,...
Construct a rational function A4 of the roots with coefficients in R such that 4 belongs to the subgroup H of G. Then
4 is a root of an equation of degree v whose group r for R is
simply isomorphic with the simple quotient group G/H. After
the adjunction of the root 4 to R, the group of the given equation becomes H for the domain (R, /).
Construct a rational function x of the roots with coefficients in (R, 4) such that x belongs to the subgroup K of H.
Then x is a root of an equation of degree p whose group for
(R, 4) is simply isomorphic with the simple group H/K. After
the adjunction of x, the group of the given equation is K.
Finally, we adjoin a function belonging to G1 and cbtain a
domain containing xi,..., x,. We therefore have
THEOREM 4. The solution of an equation with the group
G for the domain R can be reduced to the solution of a series of
equations each with a simple regular group for the domain obtained
by adjoining to R a root of each of the earlier equations of the series.
If, in particular, G is a solvable group, each auxiliary equation
has a regular cyclic group of prime order.
160. Equations with a Regular Cyclic Group. To supplement the last theorem we need the result that any equation with
a regular cyclic group of prime order p is solvable by radicals.