String duality

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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.

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