nLab Cayley's theorem

Idea

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

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).

References

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

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

• Joseph J. Rotman, Thm. 3.12 in: An Introduction to the Theory of Groups, Springer (1995) [doi:10.1007/978-1-4612-4176-8, pdf]

• James Milne, Thm. 1.22 in: Group theory (2021) [web, pdf]

See also:

• Wikipedia, Cayley’s theorem

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