Hall's universal group

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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 automorphisms of U.

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

[edit] Construction

Take any group $Γ 0$ of order $\geq 3$ . Denote by $Γ 1$ the group $S_{\Gamma_0}$ of permutations of elements of $Γ 0$ , by $Γ 2$ the group

$S_{\Gamma_1}= S_{S_{\Gamma_0}}$

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

$x\mapsto gx$

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

$\Gamma_0 \hookrightarrow \Gamma_1 \hookrightarrow \Gamma_2 \hookrightarrow ...$

A direct limit (that is, a union) of all $Γ i$ 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 $\Gamma_i \subset U$ . The group $\Gamma_{i+1}= S_{\Gamma_i}$ acts on $Γ i$ by permutations, and conjugates all possible embeddings $G \hookrightarrow U$ .