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

304

EQUATIONS SOLVABLE BY RADICALS

[CH. XV

the roots of f(x) =0 will be in Rk (property A of ~ 149). Hence
if such a series of resolvent equations can be constructed and
solved relatively to their domains, the given equation can be
solved relatively to R. Consequently, we shall discuss the
question of the solvability of a resolvent equation; to this end
we must find its group.
By use of (14) in ~ 154, we proved that P is one of v conjugate functions under the group G:
(1), p, * **v,,
and that any substitution s of G replaces these by
(2) 's, 92S), ^3S) * * ' ivs)
which are merely the distinct functions (1) rearranged. Hence
to any substitution s of G on the letters xi,..., xn there
corresponds one definite substitution
(3) a = ( * W (es)
on the v letters (1). Similarly, to t corresponds
since we may rearrange at will the letters in the upper line
in the two-rowed notation of a substitution. The product
ar replaces 'P by Pst and hence corresponds to st.
THEOREM 1. The substitutions of G correspond to substitutions (3) forming a group r.
The group r is transitive and isomorphic to G.
EXAMPLE. Let G be the alternating group on the independent variables xl,..., 4. Now 4 =(x1-x2)(x3-x4) belongs formally to the
group G4 given by (12) of ~ 150. We have
G= G4 +G4(x2x3x4) +G4(x2x4x3).
The indicated substitutions of period 3 replace f by
/2 = (X1 — 3) (X4 - X2), al3 = (Xl - X4) (X2n-X3).
Since every substitution of G4 leaves also 12 and ~/3 unaltered,
r= 11, (~~2 3), (~3~2)}S.