Cayley's theorem

In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every finite group G is isomorphic to a subgroup of the symmetric group acting 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. The set of all permutations of G 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 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".[4]

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, ${\displaystyle S_{3}}$, itself already a symmetric group of order 6, would be represented by the regular action as a subgroup of ${\displaystyle S_{6}}$ (a group of order 720).[5] The problem of finding an embedding of a group in a minimal-order symmetric group is rather difficult.[6][7]

History

While it seems elementary enough, 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[8] attributes the theorem to Jordan,[9] Eric Nummela[10] nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper,[11] 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). 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[12] and is attributed to Dyck in the first edition of Burnside's book.[13]

Proof of the theorem

If 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, ${\displaystyle f_{g^{-1}}}$. 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)):

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

for all x in G, and hence:

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

The homomorphism T is 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, i.e. the kernel is trivial. Alternatively, T is also injective since 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 ${\displaystyle G}$ as acting on itself by left multiplication, i.e. ${\displaystyle g\cdot x=gx}$, which has a permutation representation, say ${\displaystyle \phi :G\to \mathrm {Sym} (G)}$.

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

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

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 of an arbitrary group ${\displaystyle G}$. In general if ${\displaystyle G}$ is a group and ${\displaystyle H}$ is a subgroup with ${\displaystyle |G:H|<\infty }$, then ${\displaystyle G/{\text{Core}}_{G}(H)}$ is isomorphic to a subgroup of ${\displaystyle \Sigma _{|G:H|}}$. In particular if ${\displaystyle G}$ is a finite group and we set ${\displaystyle H=1}$ then we get the classic result.

See also

• Wagner–Preston theorem is the analogue for inverse semigroups.
• inclusion order, 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

1. Jacobson (2009, p. 38)
2. Jacobson (2009, p. 72, ex. 1)
3. Jacobson (2009, p. 31)
4. J. L. Alperin; Rowen B. Bell (1995). Groups and representations. Springer. p. 29. ISBN 978-0-387-94525-5.
5. Peter J. Cameron (2008). Introduction to Algebra, Second Edition. Oxford University Press. p. 134. ISBN 978-0-19-852793-0.
6. Johnson, D. L. (1971). "Minimal Permutation Representations of Finite Groups". American Journal of Mathematics. 93 (4): 857. doi:10.2307/2373739. JSTOR 2373739.
7. Grechkoseeva, M. A. (2003). "On Minimal Permutation Representations of Classical Simple Groups". Siberian Mathematical Journal. 44 (3): 443–462. doi:10.1023/A:1023860730624.
8. Burnside, William (1911), Theory of Groups of Finite Order (2 ed.), Cambridge, p. 22, ISBN 0-486-49575-2
9. Jordan, Camille (1870), Traite des substitutions et des equations algebriques, Paris: Gauther-Villars
10. Nummela, Eric (1980), "Cayley's Theorem for Topological Groups", American Mathematical Monthly, Mathematical Association of America, 87 (3): 202–203, doi:10.2307/2321608, JSTOR 2321608
11. Cayley, Arthur (1854), "On the theory of groups as depending on the symbolic equation θⁿ=1", Philosophical Magazine, 7 (42): 40–47
12. von Dyck, Walther (1882), "Gruppentheoretische Studien" [Group-theoretical Studies], Mathematische Annalen, 20 (1): 30, doi:10.1007/BF01443322, hdl:2027/njp.32101075301422, ISSN 0025-5831. (in German)
13. Burnside, William (1897), Theory of Groups of Finite Order (1 ed.), Cambridge, p. 22

References

• Jacobson, Nathan (2009), Basic algebra (2nd ed.), Dover, ISBN 978-0-486-47189-1.