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In [[algebra]], '''Hall's universal group''' is
a  countable [[locally finite group]], say ''U'', which is uniquely
characterized by the following properties.
 
* Every finite group ''G'' admits a [[monomorphism]] to ''U''.
 
* All such monomorphisms are conjugate by [[inner automorphism]]s of ''U''.
 
It was defined by [[Philip Hall]] in 1959.<ref>Hall, P.
''Some constructions for locally finite groups.''
J. London Math. Soc. '''34''' (1959) 305--319. {{MathSciNet | id = 162845}}</ref>
 
== Construction ==
 
Take any group <math> \Gamma_0 </math> of order <math> \geq 3 </math>.  
Denote by <math> \Gamma_1 </math> the group  <math> S_{\Gamma_0}</math>
of [[permutation]]s of elements of <math> \Gamma_0 </math>, by
<math>\Gamma_2 </math> the group
 
:<math> S_{\Gamma_1}= S_{S_{\Gamma_0}} \, </math>
 
and so on. Since a group acts faithfully on itself by permutations
 
:<math> x\mapsto gx \, </math>
 
according to [[Cayley's theorem]], this gives a chain of monomorphisms
 
:<math>\Gamma_0 \hookrightarrow  \Gamma_1 \hookrightarrow \Gamma_2 \hookrightarrow \cdots . \, </math>
 
A [[direct limit]] (that is, a union) of all <math> \Gamma_i</math>
is Hall's universal group ''U''.
 
Indeed, ''U'' then contains a [[symmetric group]] of arbitrarily large order, and any
group admits a monomorphism to a [[symmetric group|group of permutations]], as explained above.
Let ''G'' be a finite group admitting two embeddings to ''U''.
Since ''U'' is a direct limit and ''G'' is finite, the
images of these two embeddings belong to
<math>\Gamma_i \subset U </math>. The group
<math>\Gamma_{i+1}= S_{\Gamma_i}</math> acts on <math>\Gamma_i</math>
by permutations, and conjugates all possible embeddings
<math>G \hookrightarrow U</math>.
 
==References==
 
<references />
 
[[Category:Infinite group theory]]
[[Category:Permutation groups]]

Latest revision as of 17:46, 16 March 2013

In algebra, Hall's universal group is a countable locally finite group, say U, which is uniquely characterized by the following properties.

It was defined by Philip Hall in 1959.[1]

Construction

Take any group Γ0 of order 3. Denote by Γ1 the group SΓ0 of permutations of elements of Γ0, by Γ2 the group

SΓ1=SSΓ0

and so on. Since a group acts faithfully on itself by permutations

xgx

according to Cayley's theorem, this gives a chain of monomorphisms

Γ0Γ1Γ2.

A direct limit (that is, a union) of all Γi is Hall's universal group U.

Indeed, U then contains a symmetric group of arbitrarily large order, and any group admits a monomorphism to a group of permutations, as explained above. Let G be a finite group admitting two embeddings to U. Since U is a direct limit and G is finite, the images of these two embeddings belong to ΓiU. The group Γi+1=SΓi acts on Γi by permutations, and conjugates all possible embeddings GU.

References

  1. Hall, P. Some constructions for locally finite groups. J. London Math. Soc. 34 (1959) 305--319. Template:MathSciNet