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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.
- Every finite group G admits a monomorphism to U.
- All such monomorphisms are conjugate by inner automorphisms of U.
It was defined by Philip Hall in 1959.[1]
Construction
Take any group of order . Denote by the group of permutations of elements of , by the group
and so on. Since a group acts faithfully on itself by permutations
according to Cayley's theorem, this gives a chain of monomorphisms
A direct limit (that is, a union) of all 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 . The group acts on by permutations, and conjugates all possible embeddings .
References
- ↑ Hall, P. Some constructions for locally finite groups. J. London Math. Soc. 34 (1959) 305--319. Template:MathSciNet