Cayley's Theorem

In the mathematical discipline of group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group.
More specifically, is isomorphic to a subgroup of the symmetric group $\operatorname(G)$ whose elements are the permutations of the underlying set of .
Explicitly,
* for each $g \in G$, the left-multiplication-by- map $\ell_g \colon G \to G$ sending each element to is a permutation of , and
* the map $G \to \operatorname(G)$ sending each element to $\ell_g$ is an injective homomorphism, so it defines an isomorphism from onto a subgroup of $\operatorname(G)$.
The homomorphism $G \to \operatorname(G)$ can also be understood as arising from the left translation action of on the underlying set .

When is finite, $\operatorname(G)$ is finite too. The proof of Cayley's theorem in this case shows that if is a finite group of order , then is isomorphic to a subgroup of the standard symmetric group $S_n$. But might also be isomorphic to a subgroup of a smaller symmetric group, $S_m$ for some m; for instance, the order 6 group $G=S_3$ is not only isomorphic to a subgroup of $S_6$, but also (trivially) isomorphic to a subgroup of $S_3$. The problem of finding the minimal-order symmetric group into which a given group 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 is infinite, $\operatorname(G)$ is infinite, but Cayley's theorem still applies.

History

When Cayley (1854) introduced what are now called ''groups'', the modern definitions did not exist, and it was not immediately clear that this was equivalent to what were then called 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's original 1854 paper,
showed that the correspondence in the theorem is one-to-one, but he did not explicitly show it was a homomorphism (and thus an embedding). However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.

The theorem was later published by Walther Dyck in 1882 and is attributed to Dyck in the first edition of Burnside's book.

Background

A ''permutation'' of a set is a bijective function from to . The set of all permutations of forms a group under function composition, called ''the symmetric group on'' , and written as $\operatorname(A)$.
In particular, taking to be the underlying set of a group produces a symmetric group denoted $\operatorname(G)$.

Proof of the theorem

If ''g'' is any element of a group ''G'' with operation ∗, consider the function , defined by . By the existence of inverses, this function has also an inverse, $f_$. So multiplication by ''g'' acts as a bijective function. Thus, ''f''_''g'' is a permutation of ''G'', and so is a member of Sym(''G'').

The set is a subgroup of Sym(''G'') that is isomorphic to ''G''. The fastest way to establish this is to consider the function with for every ''g'' in ''G''. ''T'' is a group homomorphism because (using · to denote composition in Sym(''G'')):

:$(f_g \cdot f_h)(x) = f_g(f_h(x)) = f_g(h*x) = g*(h*x) = (g*h)*x = f_(x) ,$
for all ''x'' in ''G'', and hence:
:$T(g) \cdot T(h) = f_g \cdot f_h = f_ = T(g*h) .$
The homomorphism ''T'' is injective since (the identity element of Sym(''G'')) implies that for all ''x'' in ''G'', and taking ''x'' to be the identity element ''e'' of ''G'' yields , i.e. the kernel is trivial. Alternatively, ''T'' is also injective since implies that (because every group is cancellative).

Thus ''G'' is isomorphic to the image of ''T'', which is the subgroup ''K''.

''T'' is sometimes called the '' regular representation of'' ''G''.

Alternative setting of proof

An alternative setting uses the language of group actions. We consider the group $G$ as acting on itself by left multiplication, i.e. $g \cdot x = gx$, which has a permutation representation, say $\phi : G \to \mathrm(G)$.

The representation is faithful if $\phi$ is injective, that is, if the kernel of $\phi$ is trivial. Suppose $g\in\ker\phi$. Then, $g = ge = g\cdot e = e$. Thus, $\ker\phi$ is trivial. The result follows by use of the first isomorphism theorem, from which we get $\mathrm\, \phi \cong G$.

Remarks on the regular group representation

The identity element of the group corresponds to the identity permutation. All other group elements correspond to derangements: permutations that do not leave any element unchanged. Since this also applies for powers of a group element, lower than the order of that element, each element corresponds to a permutation that consists of cycles all of the same length: this length is the order of that element. The elements in each cycle form a right coset of the subgroup generated by the element.

Examples of the regular group representation

$\mathbb Z_2 = \$ with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12) (see cycle notation). E.g. 0 +1 = 1 and 1+1 = 0, so $1\mapsto0$ and $0\mapsto1,$ as they would under a permutation.

$\mathbb Z_3 = \$ with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 to permutation (123), and group element 2 to permutation (132). E.g. 1 + 1 = 2 corresponds to (123)(123) = (132).

$\mathbb Z_4 = \$ with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).

The elements of Klein four-group correspond to e, (12)(34), (13)(24), and (14)(23).

S₃ ( dihedral group of order 6) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements, and the latter is how it is realized by its regular representation.

More general statement

Theorem:
Let be a group, and let be a subgroup.
Let $G/H$ be the set of left cosets of in .
Let be the normal core of in , defined to be the intersection of the conjugates of in .
Then the quotient group $G/N$ is isomorphic to a subgroup of $\operatorname(G/H)$.

The special case $H=1$ is Cayley's original theorem.

See also

* Wagner–Preston theorem is the analogue for inverse semigroups.
* Birkhoff's representation theorem, a similar result in order theory
* Frucht's theorem, every finite group is the automorphism group of a graph
* Yoneda lemma, a generalization of Cayley's theorem in category theory
* Representation theorem

Notes

References

* {{Citation, last=Jacobson, first=Nathan, author-link=Nathan Jacobson, year=2009, title=Basic algebra, edition=2nd, publisher=Dover, isbn = 978-0-486-47189-1.

Permutations
Theorems about finite groups
Articles containing proofs