One of the first basic theorems in group theory is Cayley’s theorem, which links abstract finite groups with concrete finite groups (otherwise known as permutation groups).

Theorem 1 (Cayley’s theorem)Let $latex {G}&fg=000000$ be a group of some finite order $latex {n}&fg=000000$. Then $latex {G}&fg=000000$ is isomorphic to a subgroup $latex {tilde G}&fg=000000$ of the symmetric group $latex {S_n}&fg=000000$ on $latex {n}&fg=000000$ elements $latex {{1,dots,n}}&fg=000000$. Furthermore, this subgroup is simply transitive: given two elements $latex {x,y}&fg=000000$ of $latex {{1,dots,n}}&fg=000000$, there is precisely one element $latex {sigma}&fg=000000$ of $latex {tilde G}&fg=000000$ such that $latex {sigma(x)=y}&fg=000000$.

One can therefore think of $latex {S_n}&fg=000000$ as a sort of “universal” group that contains (up to isomorphism) all the possible groups of order $latex {n}&fg=000000$.

*Proof:* The group $latex {G}&fg=000000$ acts on itself by multiplication on the left, thus each element $latex {g in G}&fg=000000$ may be identified with a permutation $latex…

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