Left Multiplication Group Action
On this page:
Introduction
Recall that group actions formalize the idea that groups encode a permutation structure on a set. Because the left regular action allows groups to permute themselves, we can actually realize abstract groups concretely as subgroups of the symmetric group. This is known as Cayley’s theorem. As we will see, the language of group actions makes the proof of this theorem trivial. We will then develop some more theory of the left multiplication action.
Cayley’s Theorem
[theorem] [thmtitle]Theorem (Cayley’s Theorem).[/thmtitle]
Every group $G$ is isomorphic to a subgroup of some symmetric group. If $G$ is a group of order $n$, then $G$ is isomorphic to a subgroup of $\Sym_n$. [/theorem]
[proof] Let $G$ act on itself by left multiplication. Then the permutation representation of this action is the homomorphism
\[\begin{align*} G &\to S_G \\ g \mapsto \sigma_g \end{align*}\]where $\sigma_g:G \to G$ sends $x \mapsto gx$. The kernel of this homomorphism is injective because, we may take $x = 1$ which gives
\[gx = x \quad\implies\quad g = 1.\]Thus, the permutation representation restricts to an isomorphism on its image, giving the desired result. [/proof]
[example] [extitle]Remark.[/extitle] Note that Cayley’s theorem implies that it is sufficient to fully understand the symmetric group to understand any abstract group. This is not practical; however, because $\Sym_n$ contains a lot. of elements. [/example]