Difference between revisions of "Cayley's Theorem"
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We prove that each group <math>G</math> is isomorphic to a group of bijections on itself. Indeed, for all <math>g\in G</math>, let <math>f_g</math> be the mapping <math>f_g : x \mapsto gx</math> from <math>G</math> into itself. Then <math>f_g</math> is a bijection, for all <math>g</math>; and for all <math>g,h \in G</math>, <math>f_g \circ f_h = f_{gh}</math>. Thus <math>G</math> is isomorphic to the set of permutations <math>\{ f_g | g \in G\}</math> on <math>G</math>. <math>\blacksquare</math> | We prove that each group <math>G</math> is isomorphic to a group of bijections on itself. Indeed, for all <math>g\in G</math>, let <math>f_g</math> be the mapping <math>f_g : x \mapsto gx</math> from <math>G</math> into itself. Then <math>f_g</math> is a bijection, for all <math>g</math>; and for all <math>g,h \in G</math>, <math>f_g \circ f_h = f_{gh}</math>. Thus <math>G</math> is isomorphic to the set of permutations <math>\{ f_g | g \in G\}</math> on <math>G</math>. <math>\blacksquare</math> | ||
− | The action of <math>G</math> on itself as described in the proof is called the ''left action of <math>G</math> on itself''. Right action is defined similarly | + | The action of <math>G</math> on itself as described in the proof is called the ''left action of <math>G</math> on itself''. Right action is defined similarly. |
== See also == | == See also == |
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.