# Cayley's theorem. Corollary 5 Dummit and Foote

This is Corollary 5 in Dummit&Foote from chapter 4.2 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 $$|S_p| = p!$$

• Every group of order $n$ embeds in $S_n$. – Randall Aug 27 at 23:09
• Dummit & Foote has more than one author, so please say "they" rather than "he". – Shaun Aug 27 at 23:10
• Look for Cayley's Theorem. – 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 $$G/K \cong \mathrm{Im}(\pi_H) < S_p$$.
• 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 $\pi_H: G \times X \to X$ – Hawk Aug 28 at 23:24
• @Hawk Nah, $\pi_H$ isn't exactly the action. You invent $\pi_H$ based on the action. – Mike Pierce Aug 28 at 23:37