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

THE ALGEBRAIC SOLUTION OF EQUATIONS 225

Next, let Q be a normal sub-group of the alternating group,
let sl be any substitution in Q (except the substitution 1), and
s any substitution in the alternating group. It is easy to see
that, by the property of normal sub-groups, s-'sls is also a
substitution in Q.
If s, consists of a cycle of three elements, we can, by proper
selection of s in the operation s-~sls, transform s, into any
other cycle of three elements. Therefore, Q must contain all
cyclic substitutions of three elements whenever it contains one
of them, and must, consequently, be identical with the alternating group.
Since sj-1 and s-~sjs are both substitutions in Q, their product
must be; namely, X = s -1. s-~1ss.
We shall now show that, whenever n > 4, s can always be
chosen from the substitutions of the alternating group in such
a way that the substitution X represents a cycle of three elements, thereby showing that the normal sub-group Q is really
identical with the alternating group; in other words, showing
that there is no normal sub-group, distinct from the alternating
group itself, except the group 1.
To show this, we assume that all the substitutions in the
alternating group and in Q are resolved (as they always can
be) into cycles so that no two cycles have an element in comnmon, ~ 860 In the formation of X there is no need whatever
of considering those cycles in the substitutions s, whose elements are unaffected by s, because in the product ss-ls-'s, they
cancel each other. We shall consider separately the different
forms which s, may take, when n > 4.
(1) Let some one substitution st in the normal sub-group Q
have a cycle (1 2 3...-m) which consists of more than three
elements. Then s= (1 2 3 *.. m)c c * *, where c1, c, *- are cy (cles
which do not contain the elements 1 2 3) *.. m. Choose s=(1 2 3),
then s,-ls-ls, = sl-l(1 3 2) s- = (2 4 3), and X- sl-ls-'^ss (2 4 3).
(1 2 3) = (1 2 4). Hence Q contains a substitution X consisting
Q