This is Corollary 5 in Dummit&Foote from chapter 4.2
It says it is isomorphic to a subgroup of $S_p$? Why $p$? What does $H$ having $p$ cosets have to do with this?
I am trying to understand all the definitions in Dummit/Foote. Is $H \stackrel{\pi_H}\to S_H$ defined as where $\pi_H(g)(aH) = \pi_H(aH) = gaH$?
They calls a given action of $G$ affords the associated permutation representation of $G$ (page 114). Is this the same as the action? I am really confused over the terminologies.
\strike 3. They also says Lagrange's theorem imply $pk$ divides $p!$. What does Lagrage have to do with this?\strike~~
The image is a subgroup of $S_p$, with divides the order of the group $|S_p| = p!$
Let $X$ be the set of (left) cosets of $H$, so $|X|=p$. Then $G$ induces an action on $X$ via left multiplication: for a coset $xH \in X$, $G$ acts on $X$ via $g.xH = (gx)H$. With this action each $g\in G$ permutes the elements of $X$. That is, each element of $g$ corresponds to a permutation, and since $X$ has $p$ element, that permutation will live in $S_p$. This correspondence gives us a homomorphism $\pi_H \colon G \to S_p$, and the kernel of this map is $K$. By the first isomorphism theorem, we get $G/K \cong \mathrm{Im}(\pi_H) < S_p$.