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

THEORY OF EQUATIONS

tations which can be obtained from the permutation P directly
by performing one of the given substitutions. Thus it follows
that the product of any two substitutions or the square of any
substitution is equivalent to one of the given substitutions.
Ex. 1. The letters (a1a03 admit of the six permutations, ata123, ala3a2,
a2a1a3, 2ca3a1, 3aaiC2, ac32cl. Show that these six permutations are
obtained, respectively, from ala2a3 by performing the substitutions 1,
(al)(a2a3), (ala2)(a3), (alnaa3), (alada2), (Cl3s)(ao2). Show that these
substitutions form a group.
98. Symmetric Functions and Symmetric Group. A symmetric
function of n letters a,, c., *..,,, being unaltered in value when
any two of the letters are interchanged, undergoes no change
in value when it is operated on by a substitution belonging to
the group given in the preceding theorem. Because of this
invariance the symmetric function is said to belong to that group,
and the group bears the name of sylmmletric group.
Ex. 1. By applying each of the substitutions of the symmetric group
1, (ala2a3), (ala3ac), (a2a3), (aias), (alaa), show the invariance of the
symmetric function, a1a2 + a1a3 + a2a3.
99. Theorem. All even substitutions of n letters form together
a group.
Even substitutions are each resolvable into the product of an
even number of transpositions, ~ 92. Hence the product of any
two of them and the square of any one of them yield even
substitutions.
Ex. 1. With the letters a, b, c we can form three transpositions (ab),
(ac), (be). Taking the products of every two of these in either sequence
and the square of every transposition, we obtain the following distinct
substitutions, all even, which form a group:
1, (abc), (acb).
Ex. 2. Show that the odd substitutions of n letters do not form a
group.