# Difference between revisions of "Cayley's Theorem"

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− | '''Cayley's Theorem''' states that every [[group]] is [[isomorphic]] to a [[permutation group]], i.e., a [[subgroup]] of a [[symmetric group]]; in other words, every group acts on some [[set]]. Although the result is simple, it is deep, as it characterizes group structure as the structure of a family of [[bijection]]s. | + | '''Cayley's Theorem''' states that every [[group]] is [[isomorphic]] to a [[permutation group]], i.e., a [[subgroup]] of a [[symmetric group]]; in other words, every group [[group action|acts]] on some [[set]]. Although the result is simple, it is deep, as it characterizes group structure as the structure of a family of [[bijection]]s. |

=== Proof === | === Proof === | ||

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* [[Symmetric group]] | * [[Symmetric group]] | ||

+ | * [[Group action]] | ||

[[Category:Group theory]] | [[Category:Group theory]] |

## Revision as of 15:20, 7 September 2008

**Cayley's Theorem** states that every group is isomorphic to a permutation group, i.e., a subgroup of a symmetric group; in other words, every group acts on some set. Although the result is simple, it is deep, as it characterizes group structure as the structure of a family of bijections.

### Proof

We prove that each group is isomorphic to a group of bijections on itself. Indeed, for all , let be the mapping from into itself. Then is a bijection, for all ; and for all , . Thus is isomorphic to the set of permutations on .

The action of on itself as described in the proof is called the *left action of on itself*. Right action is defined similarly.