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

~ 71 POSITIVE AND NEGATIVE SUBSTITUTIONS

17

factored is either always even or it is always odd. We proceed to prove this important elementary theorem.
Let s represent any substitution on m letters and suppose
that s has been factored into transpositions in various ways.
Each of these transpositions changes the sign of the following
determinant: *
1 al2... alm-1
1 a2 a22... a2m-l
1 am am2... ammwhich is not identically zero. As the various sets of tranpositions which are equivalent to s must have the same effect
on A as s has, it results that the number of transpositions in
every set is odd if s transforms A into -A, and this number is
even if s transforms A into itself.
A substitution is said to be positive if it can be factored
into an even number of transpositions. If it can be factored
into an odd number of transpositions it is called negative. For
instance, abc = abac is positive, and abcd = ab ac ad is negative. The product of two positive substitutions is positive
and the product of two negative substitutions is also positive.
Hence a product of a set of substitutions is positive or negative
according as it involves an even or an odd number of negative
substitutions.
If a substitution group involves a negative substitution
then exactly one-half of its substitutions are negative. In
fact, if all its positive substitutions are multiplied into this
negative substitution, all of these products are distinct and
negative. Hence it has at least as many negative substitutions
as positive ones. On the other hand, if this negative substitution is multiplied into all of its negative substitutions, all
these products are distinct and positive. Hence it has at
least as many positive substitutions as negative ones. In other
* This determinant is known as the determinant of Vandermonde or of
Cauchy. It is equal to the product
II(aZ —at); i, k=l, 2,..., m; i>k,