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*. This can be understood as an example of the group action of

*G*on the elements of

*G*.

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*).

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.

### Proof of the theorem

From elementary group theory, we can see that for any element *g* in *G*, we must have *g***G* = *G*; and by cancellation rules, that *g***x* = *g***y* if and only if *x* = *y*. So multiplication by *g* acts as a bijective function *f*_{g} : *G* → *G*, by defining *f*_{g}(*x*) = *g***x*. Thus, *f*_{g} is a permutation of *G*, and so is a member of Sym(*G*).

The subset *K* of Sym(*G*) defined as *K* = {*f*_{g} : *g* in *G* and *f*_{g}(*x*) = *g***x* for all *x* in *G*} is a subgroup of Sym(*G*) which is ) = *f*_{g} • *f*_{h} = *f*_{(g*h)} = *T*(*g***h*).
The homomorphism *T* is also injective since *T*(*g*) = id_{G} (the identity element of Sym(*G*)) implies that *g*x* = *x* for all *x* in *G*, and taking *x* to be the identity element *e* of *G* yields *g* = *g***e* = *e*.

Thus *G* is isomorphic to the image of *T*, which is the subgroup *K* considered earlier.

*T* is sometimes called the *regular representation of* *G*.