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

EQUIVALENCE

263

are equivalent. If we put yl=x +X2, y2=X -X2 in S, then this transformation takes the form (1, -1) in the variables yi, y2.
THEOREM 19. Let G = (S'1, S2,..., S') and G" =(S"
S"/2, ~... S/g) be simply isomorphic linear groups in n and m
variables respectively, and let G' be transitive. Then if S"t represents the conjugate-imaginary of the transformation S"t,
9
E (S) x(S t) =kg,
t=l
where k = or a positive integer. If k = 1, G" is equivalent to G';
if k> 1, G" is intransitive, and in this case k of its sets of intransitivity are transformed according to k groups each equivalent to G'.
Proof. Let xl,..., be the variables of G' and yl,..., y those of G". We construct the group K in the nm
variables xiy,..., Xy,n, where y,..., yY are the conjugate-imaginaries of the variables of G". Applying Theorem
17 we find k linearly independent absolute invariants of K
all of the form allxiyl+... +amxnym. By a suitable
change of variables in G" we now cause one of these invariants
to become Xyl +X2y2+... +xnyn, and comparing this with
the Hermitian invariant ixlx +... +xxn of G' we may
readily prove that G" transforms the variables yi,..., y,
among themselves according to a group equivalent to G'. Hence,
if k =1 and m=n, G' and G" are equivalent; if k > 1 and m> n,
G" is reducible and therefore intransitive.
Conversely, if G" is known to break up into sets of intransitivity, k of which are transformed according to groups which
are equivalent to G', then the conjugate-imaginaries of the
variables yi,..., y of any one of these sets will combine
with xl,..., to form one invariant xiyi-+...+ xny
for K, making k such invariants in all.
COROLLARY 1. If G' and G" are equivalent, their corresponding characteristics are equal, and

if they are non-equivalent and are both transitive, this sum vanishes.