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\(\def\Z{\mathbb{Z}} \def\zn{\mathbb{Z}_n} \def\znc{\mathbb{Z}_n^\times} \def\R{\mathbb{R}} \def\Q{\mathbb{Q}} \def\C{\mathbb{C}} \def\N{\mathbb{N}} \def\M{\mathbb{M}} \def\G{\mathcal{G}} \def\0{\mathbf 0} \def\Gdot{\langle G, \cdot\,\rangle} \def\phibar{\overline{\phi}} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Ker}{Ker} \def\siml{\sim_L} \def\simr{\sim_R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

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.)


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:


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 \(S_G\text{.}\)


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.