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

226

THE LINEAR GROUPS IN TWO VARIABLES [CH. X

there is a group generated by U, and the corresponding invariant is t above. The second set contains the group generated
by S, and the corresponding invariant is the product of ~ and
I above:
W= b = xl8 +14X14X24+X28.
The third set contains the group generated by UT2, and we
get the invariant
X = X112 - 33X18X24 - 33X14X28 +-X212
These invariants satisfy the relation
108t4 - W3 + 2 = 0
(E) Icosahedral Group. We shall take the group as represented in Exercise 2, ~ 103. There are three sets of subgroups
of orders 2~, 3~ and 54, containing the groups generated by
U', S'T' and S' respectively. We get, correspondingly, the
three invariants
T = X130 +x23~ 522(X12525 - X15X225) -10005(x120X210 +X110X220),
H= -X120 - X22 + 228(X115X25 - X15X215) - 494X110X210,
f= XlX2(X110 + l llx25 - -X210),
which satisfy the relation
T2 +H3- 1728f5 =0.
EXERCISES
1. The invariant i of the tetrahedral group is the Hessian covariant
(cf. ~ 174) of the function 1:
48V\-3'= 'In1 12,
[ )21 122
and the invariant t is the Jacobian of the functions i and,:
-32/-3t= 2
I1 x2
Obtain similar relations for the octahedral and icosahedral groups.
2. From the fact that no two abelian subgroups of G can have a transformation in common (except similarity-transformations) unless they
generate a larger abelian group, it follows that G is made up of a number
of distinct abelian groups Hi, H2,..., having no transformations in