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Concept
Cayley's theorem
Summary
In 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 whose elements are the permutations of the underlying set of G.
Explicitly,
for each , the left-multiplication-by-g map sending each element x to gx is a permutation of G, and
the map sending each element g to is an injective homomorphism, so it defines an isomorphism from G onto a subgroup of .
The homomorphism can also be understood as arising from the left translation action of G on the underlying set G.
When G is finite, 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 . But G might also be isomorphic to a subgroup of a smaller symmetric group, for some ; for instance, the order 6 group is not only isomorphic to a subgroup of , but also (trivially) isomorphic to a subgroup of . The problem of finding the minimal-order symmetric group into which a given group G embeds is rather difficult.
Alperin and Bell note that "in general the fact that finite groups are imbedded in symmetric groups has not influenced the methods used to study finite groups".
When G is infinite, is infinite, but Cayley's theorem still applies.
While it seems elementary enough, at the time the modern definitions did not exist, and when Cayley introduced what are now called groups it was not immediately clear that this was equivalent to the previously known groups, which are now called permutation groups. Cayley's theorem unifies the two.
Although Burnside
attributes the theorem
to Jordan,
Eric Nummela
nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper,
showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an embedding).
Official source
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