In algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, Hall's universal group is
a countable locally finite group

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 studied....

, say U, which is uniquely
characterized by the following properties.

• Every finite group G admits a monomorphism
  Monomorphism
  In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
  
   to U.

• All such monomorphisms are conjugate by inner automorphism
  Inner automorphism
  In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
  
  s of U.

It was defined by Philip Hall

Philip Hall

Philip Hall FRS , was an English mathematician.His major work was on group theory, notably on finite groups and solvable groups.-Biography:...

 in 1959.

Construction

Take any group of order .
Denote by the group
of permutation

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

s of elements of , by
the group


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


according to Cayley's theorem

Cayley's theorem

In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...

, this gives a chain of monomorphisms


A direct limit

Direct limit

In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...

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

Indeed, U then contains a symmetric group

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

 of arbitrarily large order, and any
group admits a monomorphism to a group of permutations

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

, 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
.