This is Corollary 5 in Dummit&Foote from chapter 4.2

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  1. It says it is isomorphic to a subgroup of $S_p$? Why $p$? What does $H$ having $p$ cosets have to do with this?

  2. 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 $|S_p| = p!$

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    $\begingroup$ Every group of order $n$ embeds in $S_n$. $\endgroup$ – Randall Aug 27 at 23:09
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    $\begingroup$ Dummit & Foote has more than one author, so please say "they" rather than "he". $\endgroup$ – Shaun Aug 27 at 23:10
  • $\begingroup$ Look for Cayley's Theorem. $\endgroup$ – Qi Zhu Aug 27 at 23:14

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 $G/K \cong \mathrm{Im}(\pi_H) < S_p$.

  • $\begingroup$ igot another question. Is $\pi_H$ also the action? They way they describe $\pi_H$, makes it sound like it is a different map. But I know from the Cayley arguments that they are equivalent. Here $\pi_H: G \to S_p$, but normally actions are $\pi_H: G \times X \to X$ $\endgroup$ – Hawk Aug 28 at 23:24
  • $\begingroup$ @Hawk Nah, $\pi_H$ isn't exactly the action. You invent $\pi_H$ based on the action. $\endgroup$ – Mike Pierce Aug 28 at 23:37

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