On page 202, chapter 7, Artin makes the claim “Our first step is to note that a suitable power or x, will have prime order, say order L.
X here is an element of A_n, the alternating group, specifically a supposed normal subgroup of A_n, where x is not the identity.
Why is such a prime power order element guaranteed?
Take any $x$, the order of $x$ could be written as $L^m q$ according to the unique decomposition of integers where $L \mid x$ is a prime, $\gcd (L,q ) = 1, m \geqslant 1$. Then $$ 1 = x^{L^m q} = (x^{L^{m-1 }q}) ^L, $$ and $x^{L^{m-1}q}$ is that certain power.