An introduction to the modern theory of equations, by Florian Cajori.

SUBSTITUTION-G ROUPS

115

100. Alternating Functions and Alternating Groups. Let
al, a2, **, a(t be n magnitudes, all different. A function of
these, such that an interchange of any two of them changes
the sign of the function, is called an alternating function.
Example: (ac - a2) (a, - a3) ((t - a4)... (at - a,,)
(a2- a3) (a0 - C(4) *.. (2 - a,,)
(a,,-, an).
An even substitution performed upon this function will not
alter its value. For, an even substitution, which consists of an
even number of transpositions, will reverse the sign of the
function an even number of times, and will, therefore, restore
the function to the original sign.
Since the even substitutions of n1 letters leave an alternating
function unaltered in value while all the odd substitutions
reverse its sign, the group comprising all these even substitutions is called the alternatinr/g group of the nth degree. Because
of this invariance for all the even substitutions, but for no
others, the alternating function is said to belong to the alternating group.
* Ex. 1. Show that the square root of the discriminant of an equation
of the nth degree, expressed as a function of the roots, is a function which
belongs to the alternating group of the nth degree.
101. Cyclic Functions and Cyclic Groups. The powers of acny
substitution form a group. The number of distinct substitutions
s, s2, s3, *.., resulting from taking the different powers of the
substitution s, cannot exceed the order of the substitution (~ 87).
If this order is m, then s"1 = 1. If, therefore, we square any one
of the m distinct substitutions, or multiply any two of them
together, the result is always one of the m distinct substitutions. Hence the m distinct substitutions s, s2, s3,..., s' are a
group.