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
StatementEvery 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 Classification
Statement FormCharacterization Theorem 1 Cayley's Theorem
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Associated With
The theorem states every group embeds as a subgroup of a symmetric group, so its subject is the group concept itself.
Additional Source Wikipedia: Cayley's TheoremLead section
Has Statement Form
In Branch
Source Wikipedia: Cayley's Theorem
Named After
Principal Result Of
Source Wikipedia: Arthur Cayley
Proved By
Source Wikipedia: Cayley's Theorem
Sources
1. Wikipedia: Cayley's Theorem
Wikimedia FoundationLead section
every group G is isomorphic to a subgroup of a symmetric group
History
Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.
- In Category: Theorems
- In Branch: Algebra
- Proved By: Arthur Cayley
View the Source Wikipedia: Arthur Cayley
Wikimedia FoundationPrincipal Result Of: Arthur Cayley, Article textQuote, Principal Result Of: Arthur Cayley, Article text
In group theory, Cayley tables, Cayley graphs, and Cayley's theorem are named in his honour, as well as Cayley's formula in combinatorics.
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