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

~ 1501

TRANSITIVE GROUP

291

EXERCISES
For the domain of rational numbers, find the group of
1. X3-1=0. 2. (x-l)(x+l)(x-2)=O.
3. x3-9x+9=0 [compute (8)]. 4. x3-2=0.
5. For the domain R(c), where X is an imaginary cube root of unity,
the group of x3-2=0 is of order 3. [Compute (8)].
6. For the domain R(i), the group of x4+1=0 is of order 2.
7. Find the group G of a reciprocal quartic equation
x4+ax3+bx2+ax+1=0
for the domain R=R(a, b), when it is irreducible in R.
Hints: Choose the notation for the roots so that xLx2=x3x4=1. Then
one root of the cubic (9) is y = 2; thus y2 and y3 are the roots }b- l~VA
of y2+(2-b)y+a2-2b=O, where A-(}b+1)2-a2. The three y's are
distinct; for example, y2 -y= (xi-X4)(X3-X2). Hence G is a subgroup
of Gs, given by (10). Further, G is G4, given by (12), if and only if VA
is in R.
By the usual substitution v= x +l/x, our quartic becomes v2 av+b-2
=0, whose roots are therefore x +x2 and 3 +X4. Its discriminant
B=a2-4(b —2) is thus the square of t=x1+X2-x3-x4. Why is t0O?
Now (1324), (14)(23) and (13)(24) replace t by -t, while the first four
substitutions in (10) leave t unaltered. Again, y2-y3 is unaltered only
by the subgroup G4 of Gs, being changed in sign by the remaining four
substitutions of G8. Hence t(y2-y3) is unaltered only by the subgroup.
C4 generated by (1324). Hence G= C4 if and only if V/AB is in R.
If a transitive subgroup of G8 does not contain (1324) or its inverse
(1423), it contains (13)(24) and (14)(23), the only remaining substitutions
replacing xl by X3 and x4, respectively, and hence is G4. Thus if G is not
C4 or G4, it is G8. It follows by formal logic that G=G8 if and only if
neither VA nor V/AB is in R.
8. If the quartic in Ex. 7 is reducible in R, it is the product of two factors x2+px+r and x2+qx+l/r, where p, q, r are in R, and
P 1
p+q=a, p+rq=a, r+-+pq=b.
r r
If r=l, p and q are in R only when VB is in R. If r=-1, then a=0
and V-b-2 must be in R.i If r2Xl, we may eliminate p and q and
obtain for y=r+l/r the quadratic * in Ex. 7 with the roots y2, y3; thus R
must contain VA and the square roots of ( b-1 -/VA)2-4, the latter
being the values of (r-1/r)2=y2-4 for y=y2, y3. By Ex. 7, AB0O if
the four roots are distinct.
9. If a=b=0, then A=1, B=8, and x4tl is irreducible in R(1), and
* Except for a=; then p= q=0, y= b.