# Cayley's theorem

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In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G.[1] This can be understood as an example of the group action of G on the elements of G.[2]

A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written as Sym(G).[3]

Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. Thus, theorems which are true for permutation groups are true for groups in general.

## Contents

### History

Although Burnside[4] attributes the theorem to Jordan,[5] Eric Nummela[6] nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper,[7] showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an isomorphism). However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.

### Proof of the theorem

Where g is any element of G, consider the function fg : GG, defined by fg(x) = g*x. By the existence of inverses, this function has a two-sided inverse, $f_{g^{-1}}$. So multiplication by g acts as a bijective function. Thus, fg is a permutation of G, and so is a member of Sym(G).

The set K = {fg: g in G} is a subgroup of Sym(G) which is isomorphic to G. The fastest way to establish this is to consider the function T : G → Sym(G) with T(g) = fg for every g in G. T is a group homomorphism because (using "•" for composition in Sym(G)):