Cayley's theorem

For the number of labeled trees in graph theory, see Cayley's formula.

In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting 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 that are true for subgroups of permutation groups are true for groups in general. Nevertheless, 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".

The regular action used in the standard proof of Cayley's theorem does not produce the representation of G in a minimal-order permutation group. For example, $S_3$, itself already a symmetric group of order 6, would be represented by the regular action as a subgroup of $S_6$ (a group of order 720). The problem of finding an embedding of a group in a minimal-order symmetric group is rather more difficult.

History

While it seems elementary enough, it should be noted that at the time, the modern definitions didn't exist, and when Cayley introduced what are now called groups it wasn't 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 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 a group G with operation ∗, consider the function f_g : G → G, defined by f_g(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, f_g is a permutation of G, and so is a member of Sym(G).

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

$(f_g \cdot f_h)(x) = f_g(f_h(x)) = f_g(h*x) = g*(h*x) = (g*h)*x = f_{g*h}(x) ,$

for all x in G, and hence:

$T(g) \cdot T(h) = f_g \cdot f_h = f_{g*h} = T(g*h) .$

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. Alternatively, T is also injective since, if g ∗ x = g′ ∗ x implies that g = g′ (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 a G-set, which can be shown to have permutation representation, say $\phi$.

Firstly, suppose $G=G/H$ with $H=\{e\}$. Then the group action is $g.e$ by classification of G-orbits (also known as the orbit-stabilizer theorem).

Now, the representation is faithful if $\phi$ is injective, that is, if the kernel of $\phi$ is trivial. Suppose $g\in\ker\phi$ Then, $g=g.e=\phi(g).e$ by the equivalence of the permutation representation and the group action. But since $g \in \ker\phi$, $\phi(g)=e$ and thus $\ker\phi$ is trivial. Then $\mathrm{Im} \phi < G$ and thus the result follows by use of the first isomorphism theorem.

Remarks on the regular group representation

The identity group element corresponds to the identity permutation. All other group elements correspond to a permutation that does 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 left coset of the subgroup generated by the element.

Examples of the regular group representation

Z₂ = {0,1} with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12). E.g. 0 +1 = 1 and 1+1 = 0, so 1 -> 0 and 0 -> 1, as they would under a permutation.

Z₃ = {0,1,2} 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).

Z₄ = {0,1,2,3} with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).

The elements of Klein four-group {e, a, b, c} 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.

*	e	a	b	c	d	f	permutation
e	e	a	b	c	d	f	e
a	a	e	d	f	b	c	(12)(35)(46)
b	b	f	e	d	c	a	(13)(26)(45)
c	c	d	f	e	a	b	(14)(25)(36)
d	d	c	a	b	f	e	(156)(243)
f	f	b	c	a	e	d	(165)(234)

More general statement of the theorem

A more general statement of Cayley's theorem consist of considering the core af an arbitrary group $G$. In general if $G$ is a group and $H$ is a subgroup with $|G:H|<\infty$, then $G/Core_G(H)$ is isomorphic to a subgroup of $\Sigma_{|G:H|}$. In particular if $G$ is a finite group and we set $H=1$ then we get the classic result.

See also

• Wagner-Preston theorem is the analogue for inverse semigroups.
• Containment order, a similar result in order theory
• Frucht's theorem, every group is the automorphism group of a graph
• Yoneda lemma, an analogue of Cayley's theorem in category theory
• representation theorem

References

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