algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, Hall's universal group is a countable

locally finite group In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studie ...

, say ''U'', which is uniquely characterized by the following properties. * Every finite group ''G'' admits a

monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphis ...

to ''U''. * All such monomorphisms are conjugate by

inner automorphism In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within thos ...

s of ''U''. It was defined by

Philip Hall Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups. Biography He was educated first at Christ's Hospital, where he won the Thom ...

in 1959,Hall, P. ''Some constructions for locally finite groups.'' J. London Math. Soc. 34 (1959) 305--319. and has the universal property that ''all countable locally finite groups'' embed into it. Hall's universal group is the Fraïssé limit of the class of all finite groups.

Construction

Take any group

\Gamma_0

of order

\geq 3

. Denote by

\Gamma_1

the group

S_

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

s of elements of

\Gamma_0

, by

\Gamma_2

the group :

S_= S_ \,

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

x\mapsto gx \,

according to

Cayley's theorem In the mathematical discipline of group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group. More specifically, is isomorphic to a subgroup of the symmetric gro ...

, this gives a chain of monomorphisms :

\Gamma_0 \hookrightarrow  \Gamma_1 \hookrightarrow \Gamma_2 \hookrightarrow \cdots . \,

direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...

(that is, a union) of all

\Gamma_i

is Hall's universal group ''U''. Indeed, ''U'' then contains a

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

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_= S_{\Gamma_i}

acts on

\Gamma_i

by permutations, and conjugates all possible embeddings

G \hookrightarrow \Gamma_i

References

Infinite group theory Permutation groups