# Cayley's theorem

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 a symmetric group.[1] More specifically, G is isomorphic to a subgroup of the symmetric group {displaystyle operatorname {Sym} (G)} whose elements are the permutations of the underlying set of G. Explicitly, for each {displaystyle gin G} , the left-multiplication-by-g map {displaystyle ell _{g}colon Gto G} sending each element x to gx is a permutation of G, and the map {displaystyle Gto operatorname {Sym} (G)} sending each element g to {displaystyle ell _{g}} is an injective homomorphism, so it defines an isomorphism from G onto a subgroup of {displaystyle operatorname {Sym} (G)} .

The homomorphism {displaystyle Gto operatorname {Sym} (G)} can also be understood as arising from the left translation action of G on the underlying set G.[2] When G is finite, {displaystyle operatorname {Sym} (G)} is finite too. The proof of Cayley's theorem in this case shows that if G is a finite group of order n, then G is isomorphic to a subgroup of the standard symmetric group {displaystyle S_{n}} . But G might also be isomorphic to a subgroup of a smaller symmetric group, {displaystyle S_{m}} for some {displaystyle m 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 Theorem: Let G be a group, and let H be a subgroup. Let {displaystyle G/H} be the set of left cosets of H in G. Let N be the normal core of H in G, defined to be the intersection of the conjugates of H in G. Then the quotient group {displaystyle G/N} is isomorphic to a subgroup of {displaystyle operatorname {Sym} (G/H)} .

The special case {displaystyle 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 ^ Jacobson (2009, p. 38) ^ Jacobson (2009, p. 72, ex. 1) ^ Peter J. Cameron (2008). Introduction to Algebra, Second Edition. Oxford University Press. p. 134. ISBN 978-0-19-852793-0. ^ Johnson, D. L. (1971). "Minimal Permutation Representations of Finite Groups". American Journal of Mathematics. 93 (4): 857–866. doi:10.2307/2373739. JSTOR 2373739. ^ Grechkoseeva, M. A. (2003). "On Minimal Permutation Representations of Classical Simple Groups". Siberian Mathematical Journal. 44 (3): 443–462. doi:10.1023/A:1023860730624. S2CID 126892470. ^ J. L. Alperin; Rowen B. Bell (1995). Groups and representations. Springer. p. 29. ISBN 978-0-387-94525-5. ^ Burnside, William (1911), Theory of Groups of Finite Order (2 ed.), Cambridge, p. 22, ISBN 0-486-49575-2 ^ Jordan, Camille (1870), Traite des substitutions et des equations algebriques, Paris: Gauther-Villars ^ 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 ^ Cayley, Arthur (1854), "On the theory of groups as depending on the symbolic equation θn=1", Philosophical Magazine, 7 (42): 40–47 ^ 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, S2CID 179178038. (in German) ^ Burnside, William (1897), Theory of Groups of Finite Order (1 ed.), Cambridge, p. 22 ^ Jacobson (2009, p. 31) References Jacobson, Nathan (2009), Basic algebra (2nd ed.), Dover, ISBN 978-0-486-47189-1. Categories: PermutationsTheorems about finite groups

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