SQ-universality can be thought of as a measure of largeness or complexity of a group.
However the first explicit use of the term seems to be in an address given by Peter Neumann to The London Algebra Colloquium entitled "SQ-universal groups" on 23 May 1968.
In 1949 Graham Higman, Bernhard Neumann and Hanna Neumann proved that every countable group can be embedded in a two-generator group.
Many more examples are now known: In addition much stronger versions of the Higmann-Neumann-Neumann theorem are now known.
Ould Houcine has proved: A free group on countably many generators h1, h2, ..., hn, ... , say, must be embeddable in a quotient of an SQ-universal group G. If
This observation allows us to easily prove some elementary results about SQ-universal groups, for instance: To prove this suppose N is not SQ-universal, then there is a countable group K that cannot be embedded into a quotient group of N. Let H be any countable group, then the direct product H × K is also countable and hence can be embedded in a countable simple group S. Now, by hypothesis, G is SQ-universal so S can be embedded in a quotient group, G/M, say, of G. The second isomorphism theorem tells us: Now
Since every subgroup H of finite index in a group G contains a normal subgroup N also of finite index in G,[10] it easily follows that: Several variants of SQ-universality occur in the literature.
The reader should be warned that terminology in this area is not yet completely stable and should read this section with this caveat in mind.
is isomorphic to a subgroup of a quotient of G. The following result can be proved: Let
A group G is called SQ-universal for the class
is isomorphic to a subgroup of a quotient of G. Note that there is no requirement that
be the class of finitely presented SQ-universal groups that are G-stable for some G then Houcine's version of the HNN theorem that can be re-stated as: However, there are uncountably many finitely generated groups, and a countable group can only have countably many finitely generated subgroups.
such that F and G can be embedded in S and S can be embedded in H. The it is easy to prove: The motivation for the definition of wrappable class comes from results such as the Boone-Higman theorem, which states that a countable group G has soluble word problem if and only if it can be embedded in a simple group S that can be embedded in a finitely presented group F. Houcine has shown that the group F can be constructed so that it too has soluble word problem.
This together with the fact that taking the direct product of two groups preserves solubility of the word problem shows that: Other examples of wrappable classes of groups are: The fact that a class
It is clear, for instance, that some sort of cardinality restriction for the members of
If we replace the phrase "isomorphic to a subgroup of a quotient of" with "isomorphic to a subgroup of" in the definition of "SQ-universal", we obtain the stronger concept of S-universal (respectively S-universal for/in
The Higman Embedding Theorem can be used to prove that there is a finitely presented group that contains a copy of every finitely presented group.
is the class of all finitely presented groups with soluble word problem, then it is known that there is no uniform algorithm to solve the word problem for groups in
be the class of finitely presented groups, and F2 be the free group on two generators, we can sum this up as: The following questions are open (the second implies the first): While it is quite difficult to prove that F2 is SQ-universal, the fact that it is SQ-universal for the class of finite groups follows easily from these two facts: If
As in the group theoretic case, we use the term SQ-universal for an object that is SQ-universal both for and in the class of countable objects of
Many embedding theorems can be restated in terms of SQ-universality.
[12] However versions of the HNN theorem can be proved for categories where there is no clear idea of a free object.
[13] A similar concept holds for free lattices.
The free lattice in three generators is countably infinite.
It has, as a sublattice, the free lattice in four generators, and, by induction, as a sublattice, the free lattice in a countable number of generators.