nLab Cayley's theorem




In group theory, Cayley’s theorem is the statement that every discrete group is isomorphic to a subgroup of the symmetric group Sym(G)Sym(G) on its underlying set, namely as the group of permutations of elements of GG by left-multiplication with GG on itself (the “regular permutation representations” of GG).

Wagner-Preston theorem is a generalization of Cayley’s theorem to inverse semigroups (every inverse semigroup is a subsemigroup of a semigroup of partial bijections of a set).


Named after (cf. eg. HMM18, p. 103):

  • Arthur Cayley, On the theory of groups, as depending of the symbolic equation θ n=1\theta^n = 1. Philos.

    Mag. VII: (1854) 40–47. Reprinted in The Collected Mathematical Papers of Arthur Cayley, Vol. II., Cambridge Univeristy Press (1898)

  • Arthur Cayley, Desiderata and suggestions. No. 1. The theory of groups, Amer. J. Math. I: (1878) 50–52,

    Reprinted in: The Collected Mathematical Papers of Arthur Cayley, Vol. X, Cambridge University Prerss (1898)

Textbooks and lecture notes:

See also:

More on the case of finite groups:

  • R. Heffernan, Des Machale, Brendan McCann, Cayley’s Theorem Revisited: Embeddings of Small Finite Groups, Mathematics Magazine 91 2 (2018) 103-111 [jstor:48665514]
category: algebra

Last revised on September 4, 2023 at 17:46:35. See the history of this page for a list of all contributions to it.