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

CONJUGATE-IMAGINARY GROUPS

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where yn-1 is a linear function of xn-, Xn-2,..., xi. Continuing thus, we finally prove the theorem.
92. Conjugate-imaginary Groups and Invariant Hermitian
Form. If in a group G we replace the variables xl,..., xn
and the elements a,t of the matrices by their conjugateimaginary values xl,..., x, asn, we evidently obtain a group
G simply isomorphic with G. We shall say that either group
is the conjugate-imaginary of the other.
We say that an Hermitian form J is invariant under a group
G, or that J is an Hermitian invariant of G, when J is transformed into itself by the (intransitive) group in 2n variables
Xi,...,,, x,..., Xn made up of G and G.
THEOREM 5. There is always an invariant Hermitian form
of a given linear group G in n variables.*
Proof. Let the transformations of the group made up of
G and G be denoted by T1, T2,...., Tg, and let I represent
the Hermitian form xlxl-X2X2+... +xnx,,. Then the sum
JO= (I)T1+(I)T+ 2+... +(I)T,
is an invariant Hermitian form of G.
First, J is an Hermitian form. For, each of the terms
(I)T,a is the sum of n expressions (xxs)T,=XsXs which are
real and non-negative. The function J is therefore real and
non-negative, and cannot vanish unless every term (I) T,
vanishes. But, if T1 represents the identity, (I)T1=I and
does not vanish unless every variable xl,..., x, vanishes.
This is therefore also the case with J.
Second, J is transformed into itself by T1, T2,.., Tg.
For, evidently
(J)T = (()Tr)T,+... +((I)Tr)T,= (I)Tt+... +(I)Tr,
where
T'a = TsTa.
*This theorem was proved for n=3 by Picard and Valentiner (1887, 1889),
and for any n by Fuchs, Moore and Loewy (1896). See Encyklopddie der Mathematischen Wissenschaften, Leipzig, 1898-1904, Bd. I, 1; p. 532.