Cayley's theorem

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{{Citation | last = Burnside | first = William | author-link = William Burnside | title = Theory of Groups of Finite Order | page = 22 | location = Cambridge | year = 1911 | edition = 2 | url = https://babel.hathitrust.org/cgi/pt?id=uc1.b4062919;view=1up;seq=52;size=125 | isbn = 0-486-49575-2}}

attributes the theorem

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 | doi = 10.2307/2321608 | jstor = 2321608 | publisher = Mathematical Association of America}}

nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley's 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 = Philosophical Magazine | volume = 7 | issue = 42 | pages = 40–47 | year = 1854 | url = https://books.google.com/books?id=_LYConosISUC&pg=PA40 }}

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{{Citation | last=von Dyck | year=1882 | first=Walther | author-link=Walther Dyck | title=Gruppentheoretische Studien |trans-title=Group-theoretical Studies | url=https://archive.org/stream/mathematischean54behngoog#page/n38/mode/1up | doi=10.1007/BF01443322 | journal=Mathematische Annalen | issn=0025-5831 | volume=20 | issue=1 | page=30| hdl=2027/njp.32101075301422 | s2cid=179178038 | hdl-access=free }}. {{in lang|de}} and is attributed to Dyck in the first edition of Burnside's book.{{Citation | last = Burnside | first = William | author-link = William Burnside | title = Theory of Groups of Finite Order | page = 22 | location = Cambridge | year = 1897 | edition = 1 | url = https://archive.org/stream/cu31924086163726#page/n43/mode/2up }}

A permutation of a set {{mvar|A}} is a bijective function from {{mvar|A}} to {{mvar|A}}. The set of all permutations of {{mvar|A}} forms a group under function composition, called the symmetric group on {{mvar|A}}, and written as $\backslash operatorname\{Sym\}(A)$.{{harvtxt|Jacobson|2009|p=31}}

In particular, taking {{mvar|A}} to be the underlying set of a group {{mvar|G}} produces a symmetric group denoted $\backslash operatorname\{Sym\}(G)$.

If g is any element of a group G with operation ∗, consider the function {{nowrap|f_g : G → G}}, defined by {{nowrap|1=f_g(x) = g ∗ x}}. By the existence of inverses, this function has also an 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 {{nowrap|1=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 {{nowrap|T : G → Sym(G)}} with {{nowrap|1=T(g) = f_g}} for every g in G. T is a group homomorphism because (using · to denote composition in Sym(G)):

:$(f\_g\; \backslash cdot\; f\_h)(x)\; =\; f\_g(f\_h(x))\; =\; f\_g(h*x)\; =\; g*(h*x)\; =\; (g*h)*x\; =\; f\_\{g*h\}(x)\; ,$

for all x in G, and hence:

:$T(g)\; \backslash cdot\; T(h)\; =\; f\_g\; \backslash cdot\; f\_h\; =\; f\_\{g*h\}\; =\; T(g*h)\; .$

The homomorphism T is injective since {{nowrap|1=T(g) = id_G}} (the identity element of Sym(G)) implies that {{nowrap|1=g ∗ x = x}} for all x in G, and taking x to be the identity element e of G yields {{nowrap|1=g = g ∗ e = e}}, i.e. the kernel is trivial. Alternatively, T is also injective since {{nowrap|1=g ∗ x = g′ ∗ x}} implies that {{nowrap|1=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.

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

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

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.

$\backslash mathbb\; Z\_2\; =\; \backslash \{0,1\backslash \}$ 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 $1\backslash mapsto0$ and $0\backslash mapsto1,$ as they would under a permutation.

$\backslash mathbb\; Z\_3\; =\; \backslash \{0,1,2\backslash \}$ 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).

$\backslash mathbb\; Z\_4\; =\; \backslash \{0,1,2,3\backslash \}$ 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.

Theorem:

Let {{mvar|G}} be a group, and let {{mvar|H}} be a subgroup.

Let $G/H$ be the set of left cosets of {{mvar|H}} in {{mvar|G}}.

Let {{mvar|N}} be the normal core of {{mvar|H}} in {{mvar|G}}, defined to be the intersection of the conjugates of {{mvar|H}} in {{mvar|G}}.

Then the quotient group $G/N$ is isomorphic to a subgroup of $\backslash operatorname\{Sym\}(G/H)$.

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

• 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

{{Reflist}}

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

Category:Permutations

Category:Theorems about finite groups

Category:Articles containing proofs