In mathematics, in the realm of group theory, a countable group is said to be SQ-universal if every countable group can be embedded in one of its quotient groups. SQ-universality can be thought of as a measure of largeness or complexity of a group.
Many classic results of combinatorial group theory, going back to 1949, are now interpreted as saying that a particular group or class of groups is (are) SQ-universal. 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.[1] Using the contemporary language of SQ-universality, this result says that F2, the free group (non-abelian) on two generators, is SQ-universal. This is the first known example of an SQ-universal group. Many more examples are now known:
P=\left\langlea,b,c,d|a2=b2=c2=d2=(ab)3=(bc)3=(ac)3=(ad)3=(cd)3=(bd)3=1\right\rangle
In addition much stronger versions of the Higmann-Neumann-Neumann theorem are now known. Ould Houcine has proved:
For every countable group G there exists a 2-generator SQ-universal group H such that G can be embedded in every non-trivial quotient of H.[9]
A free group on countably many generators h1, h2, ..., hn, ..., say, must be embeddable in a quotient of an SQ-universal group G. If
| * | |
| h | |
| 2, |
...
| * | |
| ,h | |
| n |
...\inG
| * | |
| h | |
| n |
\mapstohn
Every SQ-universal group has as a subgroup, a free group on countably many generators.
Since every countable group can be embedded in a countable simple group, it is often sufficient to consider embeddings of simple groups. This observation allows us to easily prove some elementary results about SQ-universal groups, for instance:
If G is an SQ-universal group and N is a normal subgroup of G (i.e.
N\triangleleftG
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:
MN/M\congN/(M\capN)
Now
MN/M\triangleleftG/M
MN/M\capS\cong1
or:
S\subseteqMN/M\congN/(M\capN)
The latter cannot be true because it implies K ⊆ H × K ⊆ S ⊆ N/(M ∩ N) contrary to our choice of K. It follows that S can be embedded in (G/M)/(MN/M), which by the third isomorphism theorem is isomorphic to G/MN, which is in turn isomorphic to (G/N)/(MN/N). Thus S has been embedded into a quotient group of G/N, and since H ⊆ S was an arbitrary countable group, it follows that G/N is SQ-universal.
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:
If a group G is SQ-universal then so is any finite index subgroup H of G. The converse of this statement is also true.[11]
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.
Let
l{P}
l{P}
G\inl{P}
l{P}
Let n, m ∈ Z where m is odd,
n>1078
Let
l{P}
l{P}
l{P}
G\inl{P}
The standard definition of SQ-universality is equivalent to SQ-universality both in and for the class of countable groups.
Given a countable group G, call an SQ-universal group H G-stable, if every non-trivial factor group of H contains a copy of G. Let
l{G}
The free group on two generators is SQ-universal for
l{G}
However, there are uncountably many finitely generated groups, and a countable group can only have countably many finitely generated subgroups. It is easy to see from this that:
No group can be SQ-universal in
l{G}
An infinite class
l{P}
F,G\inl{P}
H\inl{P}
If
l{P}
l{P}
N\triangleleftG
l{P}
l{P}
If
l{P}
l{P}
l{P}
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:
The class of all finitely presented groups with soluble word problem is wrappable.
Other examples of wrappable classes of groups are:
The fact that a class
l{P}
l{P}
l{P}
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
l{P}
l{W}
l{W}
l{W}
l{W}
l{W}
l{F}
l{F}
l{W}
l{F}
l{W}
The following questions are open (the second implies the first):
l{W}
l{W}
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
l{C}
l{P}
l{C}
l{P}
l{C}
l{P}
l{C}
Many embedding theorems can be restated in terms of SQ-universality. Shirshov's Theorem that a Lie algebra of finite or countable dimension can be embedded into a 2-generator Lie algebra is equivalent to the statement that the 2-generator free Lie algebra is SQ-universal (in the category of Lie algebras). This can be proved by proving a version of the Higman, Neumann, Neumann theorem for Lie algebras.[12] However versions of the HNN theorem can be proved for categories where there is no clear idea of a free object. For instance it can be proved that every separable topological group is isomorphic to a topological subgroup of a group having two topological generators (that is, having a dense 2-generator subgroup).[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.[14]