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# Section6.4Cayley's Theorem

One might wonder how “common” permutation groups are in math. They are, it turns out, ubiquitous in abstract algebra: in fact, every group can be thought of as a group of permutations! We will prove this, but we first need the following lemma. (We will not use the maps $\rho_a$ or $c_a\text{,}$ defined below, in our theorem, but define them here for potential future use.)

##### Definition6.4.2

We say that $\lambda_a\text{,}$ $\rho_a\text{,}$ and $c_a$ perform on $G\text{,}$ respectively, left multiplication by $a$, right multiplication by $a$, and conjugation by $a$. (Note: Sometimes when people talk about conjugation by $a$ they instead are referring to the permutation of $G$ that sends each $x$ to $a^{-1}xa\text{.}$)

Now we are ready for our theorem:

##### Remark6.4.4

In general, $\phi(G) \neq S_G\text{,}$ so we cannot conclude that $G$ is isomorphic to $S_G$ itself; rather, we may only conclude that it is is isomorphic to some subgroup of $S_G\text{.}$

##### Remark6.4.5

While we chose to use the maps $\lambda_a$ to prove the above theorem, we could just as well have used the maps $\rho_a$ or $c_a\text{,}$ instead.