Cayley's Theorem

Definition

The left regular representation of is the Homomorphism (or Group Morphism) $σ : G \to S_{G}$ . This representation is always a Faithful Action, and transitive. It defines an injective Homomorphism (or Group Morphism). If $G$ is a finite group of order $n$ then this homomorphism establishes an Isomorphism of $G$ with a Subgroup of $S_{n}$ (or order $\leq n!$ ).

Some Corollaries

Corollary

If $G$ is a finite group of order $n$ and $p$ is the smallest prime dividing $| G |$ then any Subgroup of Index (Groups) $p$ is a Normal Subgroup.

Proof

Suppose $H \leq G$ and $| G : H | = p$ . Let $π_{H}$ be the permutation representation afforded by multiplication on the set of left cosets of $H$ in $G$ , let $K = Ker (π_{H})$ and let $| H : K | = k$ Then $| G : K | = p k$ by 32 Cosets and Lagrange's Theorem#11. Since $H$ has $p$ left cosets, then $\frac{G}{K}$ is isomorphic to a subgroup of $S_{p}$ (namely, the image of $G$ under $π_{H}$ ) by The First Isomorphism Theorem of Groups, The Fundamental Theorem for Group Morphisms. By Lagrange's Divisibility Theorem of Order of Subgroups then $p k = | \frac{G}{K} |$ divides $p!$ . Thus:

k | \frac{p!}{p} \Rightarrow k | (p - 1)!

But all the prime divisors of $(p - 1)!$ are less than $p$ and by the minimality of $p$ then every prime divisor of $k$ is greater than or equal to $p$ . This forces $k = 1$ so $H = K ⊴ G$ .

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