Cayley's theorem


Cayley's theorem

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.

History

Attributed in Burnside [Citation | last = Burnside | first = William | author-link = William Burnside | title = Theory of Groups of Finite Order | location = Cambridge | year = 1911 | edition = 2] to Jordan [Citation | last = Jordan | first = Camille | author-link = Camille Jordan | title = Traite des substitutions et des equations algebriques | publisher = Gauther-Villars | location = Paris | year = 1870] , Eric Nummela [Citation | last = Nummela | first = Eric | title = Cayley's Theorem for Topological Groups | journal = American Mathematical Monthly | volume = 87 | issue = 3 | year = 1980 | pages = 202-203] nonetheless argues that the standard name for this theorem -- "Cayley's Theorem" -- is in fact appropriate. Cayley, in his original 1854 paper [Citation | last = Cayley | first = Arthur | author-link = Arthur Cayley | title = On the theory of groups as depending on the symbolic equation θn=1 | journal = Phil. Mag. | volume = 7 | issue = 4 | pages = 40-47 | year = 1854] which introduced the concept of a group, 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

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 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 "•" for composition in Sym("G")):("f""g" • "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") • "T"("h") = "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". Alternatively, "T"("g") is also injective since, if "g"*"x"="g"*"x' " implies "x"="x' " (by pre-multiplying with the inverse of "g", which exists because "G" is a group).

Thus "G" is isomorphic to the image of "T", which is the subgroup "K".

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

Alternate setting of proof

An alternate 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 ∈ ker phi Then, g=g.e=phi(g).e by the equivalence of the permutation representation and the group action. But since g ∈ ker phi, phi(g)=e and thus ker phi is trivial. Then 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 which consists of cycles which are 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).

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:

See also

*Yoneda lemma

References


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Cayley–Hamilton theorem — In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over the real or complex field satisfies its own characteristic equation.More precisely; if A is… …   Wikipedia

  • Cayley-Bacharach theorem — In mathematics, the Cayley Bacharach theorem is a statement in projective geometry which contains as a special case Pascal s theorem. The Cayley Bacharach theorem pertains to the family of cubic curves (plane curves of degree three) passing… …   Wikipedia

  • Cayley graph — In mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a discrete group. Its definition is suggested by Cayley s theorem (named after Arthur Cayley) and uses a particular, usually… …   Wikipedia

  • Cayley-Hamilton — Théorème de Cayley Hamilton Pour les articles homonymes, voir Hamilton. En algèbre linéaire, le théorème de Cayley Hamilton (qui porte les noms des mathématiciens Arthur Cayley et William Hamilton) affirme que tout endomorphisme d un espace… …   Wikipédia en Français

  • Cayley's formula — 2^{2 2}=1 tree with 2 vertices,3^{3 2}=3 trees with 3 vertices and 4^{4 2}=16trees with 4 vertices.In mathematics, Cayley s formula is a result in graph theory named after Arthur Cayley. It states that if n is an integer bigger than 1, the number …   Wikipedia

  • Cayley plane — In mathematics, the Cayley plane (or octonionic projective plane) OP2 is a projective plane over the octonions.[1] It was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley (for his 1845 paper describing the octonions). As a… …   Wikipedia

  • Arthur Cayley — Infobox Scientist name = Arthur Cayley |242px image width = 242px caption = Portrait in London by Barraud Jerrard birth date = birth date|1821|8|16|mf=y birth place = Richmond, Surrey, UK residence = England nationality = British death date =… …   Wikipedia

  • Arthur Cayley — (* 16. August 1821 in Richmond upon Thames, Surrey; † 26. Januar 1895 in Cambridge) war ein englischer Mathematiker. Er befasste sich mit sehr vielen Gebieten der Mathematik von der Analysis, Algebra, Geo …   Deutsch Wikipedia

  • Theoreme de Cayley-Hamilton — Théorème de Cayley Hamilton Pour les articles homonymes, voir Hamilton. En algèbre linéaire, le théorème de Cayley Hamilton (qui porte les noms des mathématiciens Arthur Cayley et William Hamilton) affirme que tout endomorphisme d un espace… …   Wikipédia en Français

  • Théorème de Hamilton-Cayley — Théorème de Cayley Hamilton Pour les articles homonymes, voir Hamilton. En algèbre linéaire, le théorème de Cayley Hamilton (qui porte les noms des mathématiciens Arthur Cayley et William Hamilton) affirme que tout endomorphisme d un espace… …   Wikipédia en Français