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

262

GROUP CHARACTERISTICS

[CH. XIII

The characteristic of S' is the product of the characteristics
of S and S: x(S') = x(S)x(S) (this is seen readily when S (and
therefore S and S') is written in canonical form). Hence, by
Theorem 17, the number of linearly independent absolute
invariants of the first degree in the variables of K is tXt[/g.
Any such invariant can be thrown into the form
f=Xlxl+X2x2+... +XnXn,
where X1,.., X, are linear functions of xi,..., x.
We know one such invariant already, namely the Hermitian
invariant (~ 92), and we may assume the variables originally
so chosen in G and G that this invariant is
I=XlXl+X2X2+.. +XnXn.
Then, if X be any constant, the expression
f+\I=(X1+Xxl)xl+(X2+XX2)X2+.. +(Xn+Xn)Xn
is also an invariant.
Now, the constant X may always be determined such that
Xi+X1, X2+Xx2,..., Xn+-xn are not linearly independent.
Therefore either G is intransitive by the lemma above, or
f+XI vanishes identically. Hence, since the first alternative
violates the assumption of the theorem, any invariant f of
K is merely a constant multiple of I (viz., f= -I); in other
words, the number Extxt/g of linearly independent invariants
f is unity. The theorem follows.
EXERCISE
Prove that if G is intransitive, EXtXt=lg, where I is a positive integer
greater than 1.
131. Equivalence. Two simply isomorphic groups are
equivalent if a suitable change of variables in one will make
the matrices of their corresponding transformations identical.
If no such choice of variables is possible, the groups are nonequivalent.
For example, the groups generated by the transformations
= 0, T=(1, -1)
1 0