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

~ 1051

THE FUNDAMENTAL INVARIANTS

227

common except E and El. Hence, if the orders of these groups be respectively hi,..., we must have
(1) g~=2 *+(hl~-2 *)+(h2-2)+..
Now, h,... are factors of g, say hl=g/gl,..., and there are either
gi/2 or gi subgroups conjugate to Hi, according as there is or is not a transformation in G which permutes the linear-invariants of Hi. Hence, adding
the corresponding terms in the right-hand member of (1), we obtain
g,= 2+ E gi(hl4,-2)+ E g1(h0-2)
=2 E (g - 2g) + E (g - 2g'),
or
1 1
( h) i ( )
Verify this (Diophantine) equation for the groups (A) to (E).
3. Prove that, in the case of (D) or (E), any invariant of degree g, say
J, is an absolute invariant; that is, it is transformed into itself by every
transformation of G.
Prove also that J is a rational integral function of two of the three
fundamental invariants listed above for the respective group.
* Counting the transformations E and E1 once each.