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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>
  
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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 [https://artofproblemsolving.com/wiki/index.php/TOTO_SLOT_:_SITUS_TOTO_SLOT_MAXWIN_TERBAIK_DAN_TERPERCAYA TOTO SLOT].
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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 $G$ is isomorphic to a group of bijections on itself. Indeed, for all $g\in G$, let $f_g$ be the mapping $f_g : x \mapsto gx$ from $G$ into itself. Then $f_g$ is a bijection, for all $g$; and for all $g,h \in G$, $f_g \circ f_h = f_{gh}$. Thus $G$ is isomorphic to the set of permutations $\{ f_g | g \in G\}$ on $G$. $\blacksquare$

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

See also

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