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]] faithfully 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]] | ||
| + | [[Category: Theorems]] | ||
Latest revision as of 08:12, 29 August 2024
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 faithfully 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.
