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

SUBST ITUTION-GROUPS

125

We have seen that the alternating group, regarded as a subgroup of the symmetric group, has the alternating function
which belongs to it (~ 100). Similarly the cyclic group, regarded as a sub-group of the symmetric group, has the cyclic
function which belonys to it (~ 101). The cyclic function still
belongs to the cyclic group when the latter is considered as a
sub-group of a sub-group of the symmetric group.
The function x + x - x, - x4 belongs to the group 1, (1 3)j
(2 4) when this group is taken as a sub-group of 1, (1 3) (2 4),
(1 2)(3 4), (1 4)(2 3), but the function no longer belongs to
that group when considered as a sib-group of the symmetric
group; for the substitution (1 3) occurs in the symmetric group,
but not in the given sub-group, and yet (1 3) leaves the function unchanged. When we say that a function belongs to a
group, but do not mention of what other group the given group is
a sub-group, we shall understand that it is under the symmetric.
112. To find Functions which belong to a Group. Let G, be
a sub-group of G, G being of the degree it, and let ca, a2,..., r,
be distinct quantities. Let also
P: f:= ((1, * ' )
be a rational function which may have rational coefficients and
which will assume a different value for every substitution of
the group G. If the order of the sub-group Gl is m, we obtain,
on operating upon p with the substitutions in G,, n, distinct
values, P, Pt, P2, *'* P,.-.1 I
If now we operate upon the functions I by- any substitution
in G1, these quantities are merely permuted among themselves;
for, any value p' thus obtained as the result of two substitutions, sa and ss, of the sub-group G,, is the same as that obtained
from p by the simple substitution, s. = s,,, of this sub-group.
These facts point to the unexpected conclusion that, in the theory under
development, the equation /(.x) = m) ay 1elreesent a more general case when
the coefficients are parti(ular mnmbers than when they alr variables. See ~ 2.