Cayley's theorem

In the mathematical discipline of group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group. More specifically, G is isomorphic to a subgroup of the symmetric group Sym ⁡ ( G ) {\displaystyle \operatorname {Sym} (G)} whose elements are the permutations of the underlying set of G. Explicitly, for each g ∈ G {\displaystyle g\in G} , the left-multiplication-by-g map ℓ g : G → G {\displaystyle \ell _{g}\colon G\to G} sending each element x to gx is a permutation of G, and the map G → Sym ⁡ ( G ) {\displaystyle G\to \operatorname {Sym} (G)} sending each element g to ℓ g {\displaystyle \ell _{g}} is an injective homomorphism, so it defines an isomorphism from G onto a subgroup of Sym ⁡ ( G ) {\displaystyle \operatorname {Sym} (G)} . The homomorphism G → Sym ⁡ ( G ) {\displaystyle G\to \operatorname {Sym} (G)} can also be understood as arising from the left translation action of G on the underlying set G. When G is finite, Sym ⁡ ( G ) {\displaystyle \operatorname {Sym} (G)} is finite too. The proof of Cayley's theorem in this case shows that if G is a finite group of order n, then G is isomorphic to a subgroup of the standard symmetric group S n {\displaystyle S_{n}} . But G might also be isomorphic to a subgroup of a smaller symmetric group, S m {\displaystyle S_{m}} for some m < n {\displaystyle m<n} ; for instance, the order 6 group G = S 3 {\displaystyle G=S_{3}} is not only isomorphic to a subgroup of S 6 {\displaystyle S_{6}} , but also (trivially) isomorphic to a subgroup of S 3 {\displaystyle S_{3}} . The problem of finding the minimal-order...