It's said that most groups arise through their actions. For instance, Galois groups arise in Galois theory as automorphisms of field extensions. Linear groups arise as automorphisms of vector spaces, permutation groups arise as automorphisms of sets, and so on.
On the other hand, abelian groups often arise without any natural (or at least obvious) action — the "class groups" such as the ideal class group and Picard group, as well as the various homology groups and higher homotopy groups in topology are examples. [ADDED: One (sloppy?) way of putting it is that abelian groups arise quite often for "bookkeeping" purposes, where we think of them simply as more efficient ways to store invariants, and their actions are not obvious and not necessary for most of their basic applications.]
What are some good examples of non-abelian groups that arise without any natural action? Or, where the way the group is defined doesn't seem to indicate any natural action at all, even though there may be an action lurking somewhere? The only prima facie example I could think of was the fundamental group of a topological space, but as we know from covering space theory, for nice enough spaces (locally path-connected and semilocally simply connected), the fundamental group is the group of deck transformations on the universal covering space.
This might be somewhat related to the question raised here: Why do groups and abelian groups feel so different?.
To clarify: There are surely a lot of ways of constructing groups within group theory (or using the tools of group theory, which includes various kinds of semidirect and free products, presentations, etc.) where there is no natural action. These examples are of interest, but what I'm most interested in is cases where such groups seem to arise fully formed from something that's not group theory, and there is at least no immediate way of seeing an action of the group that illuminates what's happening.
Best Answer
Here are some examples.
Coverings of matrix groups
$SL_2(R)$ acts naturally on the plane, but its universal cover is not a matrix group, and there is no obvious natural action you can use to define it.
Units in fields and algebras
The set of quaternions of norm one is a non-abelian group that is defined without reference to a specific action. It can be identified with $SU_2(C)$ but the identification is made through a number of non canonical choices. More generally, spin groups used in physics arise naturally from Clifford algebras.
Groups defined using generators and relations
Braid groups, the Baumslag-Solitar group (which admits no faithful finite dimensional representation), are usually defined that way. Building actions of these groups on some geometric space (e.g. on the associated Cayley graph) is a way to understand these groups, but this is not the only one. This is the subject of geometric group theory.