Is there a ground between Set Theory and Group Theory/Algebra? It is well known that there are strong links between Set Theory and Topology/Real Analysis.
For instance, the study of Suslin's Problem turns out to be a set theoretic problem, even though it started in topology: namely, whether $\mathbb{R}$ is the only complete dense unbounded linearly ordered set that satisfies the c.c.c.
Another instance is when we see that what's behind extending Lebesgue Measure is really the theory of large cardinals, with the introduction of measurable cardinals. Also another example of a real analysis problem that ends up in Set Theory is whether every set of reals is measurable. So the links are clear between Set Theory and Topology/Real analysis.
My question is this: are there links, as strong as the ones I roughly described in the last paragraph, between Set Theory and Abstract Algebra? The only example I know of is the Set Theoretic solution to the famous Whitehead Problem by Shelah (namely that if $V=L$ then every Whitehead group is free and if MA+$\neg$CH then there is a Whitehead group which is not free).
Can we hope to discover more of these type of links between Set Theory and Abstract Algebra? In contrast, Model Theory seems to be strongly grounded in Abstract algebra. I have seen that Shelah has some papers about uncountable free Abelian groups but he seems to be the only one investigating some areas of Abstract Algebra with the help of Set Theory. So again is there hope for links?
 A: It turns out that when it comes to infinite groups/modules, some algebraic concepts are deeply connected to the underlying set theory (for example, the notion of freeness, the structure of Ext, etc).
A good reference for this subject is the book "Almost free modules"  by Eklof and Mekler. This book introduces the works of Shelah, Gobel, Eklof and many other important contributors in this field. This research has also led to some interesting developments in "pure" set theory, such as the introduction of black-boxes by Shelah (some diamond-like combinatorial principles which can be proved in ZFC alone, and allow the construction of many interesting algebraic objects).
A: Shelah is definitely not alone! Here are few set theorists who have done substantial algebraic work.


*

*Andreas Blass

*Paul Eklof

*Vladimir Pestov

*Simon Thomas
A: You may want to determine which discipline of set theory to link to another discipline.  Descriptive Set Theory came (roughly) as a result of foundational issues arising from looking at certain arguments in Topology and Real Analysis, and at some point later ties to Model Theory, Proof Theory, and Recursion Theory were also investigated.
If you look at results in Universal Algebra, you will find many links to various foundational disciplines.  This is probably the easiest source to find the kinds of links you mention.  For example, as a weak parallel to Shelah's Classification Theory, one finds looking at varieties of algebras and considering their spectra, and classifying those which have many models in algebraic terms to those which have few models.  People such as Baldwin, Jeong, Jezek, Kearnes, McKenzie, Valeriote, and Wood do work on decidability, the lattice of interpretability types, spectra, and other questions in the context of varieties.
There is also algebraic logic, cylindric algebras, and algebras being used to study certain aspects of set theory and logic.  You may want to peruse some of that material and then revisit the question of what links you would still like to see.
Gerhard "Ask Me About System Design" Paseman, 2010.04.06
A: The subject of Borel Equivalence relation theory involves deep connections between set theory, particularly descriptive set theory, and classification problems in algebra. The principal theme of the subject is to investigate the complexity of various naturally-occuring equivalence relations, such as the isomorphism relation on finitely generated groups, which arise in other areas of mathematics. It turns out that many of these relations can be viewed as Borel relations on a standard Borel space, and they fit into a hierarchy under the concept of Borel reducibility, introduced by Harvey Friedman. I explained a little about the subject in this MO answer. 
Much of the best work in this subject is characterized by deep connections between set theory and algebra. 
A: Descriptive set theory also has something to say about algebra ... For example, the Higman-Neumann-Neumann Embedding Theorem
states that any countable group G can be embedded into a
2-generator group K. In the standard proof of this classical
theorem, the construction of the group K involves an
enumeration of a set of generators of the group G; and it is
clear that the isomorphism type of K usually depends upon
both the generating set and the particular enumeration
that is used. So it is natural to ask whether there is a more uniform construction with the property that the
isomorphism type of K only depends upon the isomorphism
type of G. As if ...
Assume the existence of a Ramsey cardinal and
suppose that G |----> F(G) is a Borel map from the space of countable groups to the space of finitely generated groups such that G embeds into F(G). Then there exists
an uncountable set of pairwise isomorphic groups G such that
the f.g. groups F(G) are pairwise incomparable with respect
to relative constructibility; ie while G, H are isomorphic,
F(G) doesn't even lie in the "set-theoretic universe generated
by F(H)."
A: A curious example is the linear ordering on braid groups, first
discovered by Patrick Dehornoy as a consequence of a large cardinal 
axiom. Proofs without set theory were discovered later -- and also earlier,
but unpublished, by Thurston -- but Dehornoy
believes that intuition from set theory was crucial in his discovery
of the result. See his book Braids and Self Distributivity, particularly
the Introduction, which is available here.
A: Let me also mention some of the work that's been done on the automorphism tower problem in group theory. This topic was the mentioned in this early classic MO question (from before my time here). The basic situation is that one starts with a group G, and then iteratively computes the automorphism group Aut(G), the automorphism group of THAT group, and so on. The automorphism tower can be continued transfinitely, by taking a direct limit of the natural system of homomorphisms mapping a group element to the corresponding inner automorphism. When the group G is centerless, then every group in the tower is centerless, and the groups can be viewed as growing. The main question is: does the tower ever stop growing? Does one reach a fixed point? Simon Thomas proved that the answer is yes for centerless groups and I proved yes for all groups, by showing that every group leads eventually to a centerless group. 
But the connections with set theory become very interesting. Simon Thomas and I proved that there can be a group G, whose automorphism tower depends highly on the set-theoretic background, in the sense that there are forcing extensions of the universe in which the very same group G has towers of different height.  We can make the tower taller or shorter, as desired. The point is that even if one has the same group G, then the the automorphism group Aut(G) already depends on the set-theoretic background, since one can sometimes add new generic automorphisms by forcing. For example, it is sometimes possible for a complete group (centerless + no outer automorphisms) to gain new outer automorphisms in a forcing extension. This would be an example of a tower increasing from height 0 to height at least 1 (and it might still grow much taller!). This phenomenon has now been extended by Gunter Fuchs and Philipp Lücke, who showed that almost any successive up-down pattern is achievable in subsequent forcing extensions, by iterated forcing. The general conclusion is that the automorphism tower of a group, even a finite group, exhibits a fundamentally set-theoretic nature, akin to iteratively computing the power set.
Simon Thomas, who I see has posted an answer to this question, is currently writing a book on the automorphism tower problem, and it is excellent, from the preliminary versions I've seen.
A: The following older MO question is pretty relevant: What are some reasonable-sounding statements that are independent of ZFC?
In particular, the highest voted answer (by Daniel Erman) is truly mindboggling: 
Here's an example from commutative algebra. The projective dimension of a module M is defined as the minimal length of a projective resolution of M. Let S be the ring ℂ[x,y,z] and M be the module ℂ(x,y,z). Then the projective dimension of M is undecidable in ZFC. More specifically, the projective dimension of M is 2 if the continuum hypothesis holds, and it is 3 if the continuum hypothesis fails.
