Theorems

Cayley's Theorem

KAY-lee; named for Arthur Cayley

Algebra

Citation Formats

General Reference

Mathematics Atlas. "Cayley's Theorem." Accessed August 30, 2026. https://dev-mathematics.interactiveatlas.org/theorems/cayleys-theorem.

APA Style

Mathematics Atlas. (n.d.). Cayley's Theorem. Retrieved August 30, 2026, from https://dev-mathematics.interactiveatlas.org/theorems/cayleys-theorem

BibTeX

@misc{mathematicsatlas-cayley-s-theorem,
  author = {Mathematics Atlas},
  title = {Cayley's Theorem},
  year = {2026},
  url = {https://dev-mathematics.interactiveatlas.org/theorems/cayleys-theorem},
  note = {Accessed August 30, 2026}
}

A foundational result of abstract group theory showing that every group, however it is defined, can be viewed concretely as a group of permutations. Arthur Cayley stated the correspondence in an 1854 paper but did not explicitly show it was a structure-preserving embedding rather than a mere one-to-one correspondence, so the theorem predates general recognition of its own complete modern proof by decades.

Facts

Statement

Every group G is isomorphic to a subgroup of a symmetric group; concretely, G embeds into the group of permutations of its own underlying set. 1

Proof Year

1854 1

Cross-Tradition Connections