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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.

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What about unit groups of rings, etc? Though they act as automorphisms on the ring. But any group has an action if you search hard enough :) – Steve D May 18 '10 at 14:13
What about fundamental groups? – Xandi Tuni May 18 '10 at 14:27
Vipul mentions fundamental groups in his question - he rules them out because of deck transformations. – Steven Gubkin May 18 '10 at 14:45
Of course, your class group example is the same as your fundamental group example -- you can get the action (via the isomorphism with the Galois group of the Hilbert class field), but only after you think a harder. – Cam McLeman May 18 '10 at 16:30
Shouldn't this question be Community Wiki? – David White Jul 19 '11 at 2:52

10 Answers 10

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.

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The braid group arises "in nature" as a fundamental group, though. – Qiaochu Yuan May 18 '10 at 15:34
@Qiaochu: The group of unit quaternions is not isomorphic to its image under the adjoint action, because it has a nontrivial center. You could instead use the left regular action on the quaternions. – S. Carnahan May 18 '10 at 16:21
The lamplighter group is named by its (free) action on a person (the lamplighter) and a bunch of streetlamps! – Ben Wieland May 18 '10 at 18:14
It seems to me that every group arises with an action, via Cayley's Theorem, regardless of how it might be defined. – James May 18 '10 at 20:12
I agree with James. The question is about actions the OP doesn't feel are "natural"; as such, it is quite subjective, and I don't see how there could be a "correct" answer. – Steve D May 18 '10 at 23:17

The Weil group is an extension of the absolute Galois group of a number field by the connected component of the identity of its idele class group. Of course, the quotient given by the Galois group acts on stuff. Whether the bigger group naturally acts on some space is a big open problem in number theory that some people think holds the key to the Riemann hypothesis.

Tate, J. Number Theoretic Background, Proc. Symp. Pure Math. 33 (1979) 3-26.

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$E_8$, before string theory.

The Monster, before the Moonshine Module, or at least before the Griess algebra.

The 3-connected group that maps to a compact simple Lie group $K$, inducing isomorphisms on $\pi_k$ for $k>3$. (AKA the "string group".)

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It's a little debatable whether the Monster arises "naturally." – Qiaochu Yuan May 19 '10 at 3:54
Even before string theory, $E_8$ was recognized as the group of automorphisms of a $3$-form on a vector space of dimension 248. Moreover, although he didn't say this explicitly, one can combine remarks in Cartan's thesis to exhibit $E_8$ as the group of contact transformations preserving a certain first order PDE for $1$ function of $28$ variables. (I'll grant, though, that these are obscure.) – Robert Bryant Jul 18 '11 at 19:12
As I recall the wreathed biMonster arises somewhat naturally: – Allen Knutson Jul 18 '11 at 21:09
Does something really count as "showing up naturally" if the first time anyone was led to consider it was in the process of trying to exhaustively classify all objects of its type? – Timothy Chow Jul 19 '11 at 0:27
@Robert: of course one can prove that any reductive group is determined by a finite number of invariant tensors, and then by happenstance a rank 3 tensor turns out to be enough to capture $E_8$. On the Monster debatability, see… – Allen Knutson Jul 20 '11 at 4:24

I'll describe some general strategies for constructing (nonabelian) groups without referring to them as symmetries of something. Whether the examples arise "in nature" or certain actions are "natural" is sometimes debatable - this is an ambiguity in the question that may be difficult to remove. I think for the purposes of this discussion, regular representations should not qualify as "natural" actions, even though they are quite natural.

Extensions of other groups: Given two groups $H$ and $K$, pick some group that fits into an exact sequence $1 \to H \to G \to K \to 1$ by specifying some datum, homological or otherwise. You might claim that $G$ acts on the total space of a certain $H$-torsor over $K$, or on some induced representation of $H$, but this seems to come close to regular representations. Anyway, examples include:

  1. Central extensions of linear groups, suggested by coudy. The "natural" ones often have natural actions on infinite dimensional spaces (cf. Weil representation), so picking out an example that satisfies the conditions of the question could be thorny.
  2. String groups (3-connected extensions of compact simple Lie groups by $K(\mathbb{Z},3)$), suggested by Allen. You could reasonably claim that these groups naturally act on categories appearing in the WZW model, such as module categories of certain vertex algebras, but it is not obvious if you only saw a purely topological construction.
  3. Random finite $p$-groups, e.g., constructed by taking field-valued points on iterated extensions of finite length Witt vector groups. Most finite groups seem to have this form.
  4. Finite perfect groups - not very well understood outside simple groups and their central extensions.

Quotients of large groups by normal subgroups: Taking a quotient tends to destroy an action. Examples:

  1. Take a free group and start tossing in relations. Yes, this is group of symmetries of a certain 2-complex, but that 2-complex is built out of the regular representation. I mentioned groups defined by presentations and automatic groups in a previous version of this answer, and they fit in here nicely.
  2. Take a higher-categorical group and consider its $\pi_0$. The invertible objects in a monoidal category $\mathcal{C}$ form a 2-group, and their isomorphism classes form the Picard group. Mariano mentioned in the comments that this group acts naturally on the category in a weak sense, but the honest symmetries are given as a central extension of the Picard group by $B(\operatorname{Aut} 1)$

Intrinsic properties: I don't have a good example of this, but in principle, there could be a group in nature that was uniquely defined by some property, but didn't have a natural action arising from that property. One could argue that some of the finite simple groups constructed in the classification program were "found" by searching through possible centralizers of involutions and deducing consequent properties, but in the end, almost all of the groups were explicitly constructed by viewing them as symmetry groups of combinatorial or linear-algebraic objects. One could argue that some of the constructions given computationally by explicit generating matrices (e.g., some Janko groups) are unnatural, and I might agree.

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I'd say that the Picard group of a monoidal category acts on the category itself quite naturally (when the category is strict, at least; otherwise you get an action "up to" something appropriate, but an action nonetheless) – Mariano Suárez-Alvarez May 18 '10 at 21:27
An example of a group defined by an intrinsic property is a free group defined by the universal mapping property. Actually, any group defined by means of a universal mapping property might work here. I was hesitant to offer this as an answer, because one then has the obligation to construct the beast. At that point, we get into the calculus of words and, in order to get associativity, the first thing you want to do is ... – James May 19 '10 at 15:25
Scott, what lies behind your remark that random finite p-groups arise as field-valued points of finite length p-typical Witt vectors? – KConrad May 20 '10 at 6:04
@KConrad, I meant to say that for each finite p-group $G$, one could cook up a connected unipotent algebraic group $U$ so that $G \cong U(\mathbb{F}_p)$. I don't have a very good reason for including the p-typical Witt vectors; I think I mentioned them because my first exposure to unipotent groups didn't include them. – S. Carnahan May 20 '10 at 14:03

I'm surprised that nobody has mentioned the Steinberg group yet:

It is the universal central extension of $SL_n(k)$, where $k$ is a field (in greatest generality, $k$ is allowed to be a non-commutative ring, but then $SL_n(k)$ no longer makes sense). It can also be defined by generators and relations: Take the elementary matrices (matrices that differ from the identity in exactly one off-diagonal spot) and write down all the "obvious relations" that they satisfy in $SL_n(k)$. Unless you're very astute, your group will fail to be $SL_n(k)$: it will be the Steinberg group instead. The kernel of the natural map from $St_n(k) \to SL_n(k)$ is the second algebraic $K$-theory group of $k$.

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But that acts on $k^n$ in a quite natural way! – Mariano Suárez-Alvarez Jul 19 '11 at 2:38
Not faithful though (every group has a "natural" action on the one point set). – André Henriques Jul 19 '11 at 13:38

What about groups with unsolvable word problem? These were originally constructed, independently by Novikov and Boone, using finite presentations derived ultimately from codifications of Turing machines with unsolvable halting problems (It is relatively easy to get semigroups with unsolvable word problem in this way. The hard part of the construction involves the use of HNN-extensions to convert them into group presentations.) It is difficult to envisage any natural associated action of such groups.

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Higman's group seems like a pretty good example. Of course, it acts on itself, but it has no action on any finite set or finite dimensional vector space, and the only reasonable description is by generators and relations.

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I'm not sure if this counts as arising fully formed from something that's not group theory... – Qiaochu Yuan Jul 19 '11 at 2:51

Diagram groups and picture groups arise as groups of certain pictures. Also see the braided versions of Thompson's group introduced by Brin and Dehornoy.

Answering the question below, diagram groups are directed homotopy groups of directed 2-complexes (2-categories enriched over groupoids). See: Guba, V. S.; Sapir, M. V. Diagram groups and directed 2-complexes: homotopy and homology. J. Pure Appl. Algebra 205 (2006), no. 1, 1–47.

Unlike ordinary homotopy groups, these are typically non-abelian. Examples are the free groups, free Abelian groups, R. Thompson group $F$ (which is the diagram group of the dunce hat viewed as a directed 2-complex), and its relatives, as well as iterative wreath products of integers, and many other groups. Braided picture groups are defined similarly, examples are the simple R. Thompson group $V$ (again corresponds to the dunce hat). These groups do act on nice CAT(0) cubical complexes (see Farley's paper above) but this is not how they are defined. About the groups introduced by Brin and Dehornoy see the paper above.

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Dear Mark, your post would be more useful if it included a few words about those (classes of) groups that you mention. – André Henriques Jul 19 '11 at 13:59
@Andre: There are links to papers where these are defined (the first one is a Memoirs book). – Mark Sapir Jul 19 '11 at 14:12

Representation groups are a nice example. If $G$ is a finite group of order $n$ and if $m$ is the order of $H^2(G,\mathbb{C}^*)$, then a representation group is a group written as a central extension $$1\rightarrow A\rightarrow H\rightarrow G\rightarrow 1$$ such that $H$ has order $mn$ and every projective representation of $G$ lifts to a linear representation of $H$. It is a fact that representation groups always exist, although they are not unique. For instance, both the dihedral group $D_8$ of order $8$ and the quaternion group $Q_8$ are representation groups for $(\mathbb{Z}/2)^2$. I feel like these are particularly relevant because if $G$ is an abelian $p$-group with at least two cyclic factors, then any such $H$ must be non-abelian.

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In Generalized Musical Intervals and Transformations, chapter 4, Dawid Lewin describes a group of rhythmic intervals with the group operation $(i,p)*(j,q) = (i + pj,pq)$, where $i,j,p$ and $q$ are rationals.

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That's the group of affine transformations of the rationals: (i,p) sends x to px+i, so (i,p)*(j,q) sends x to p(qx+j)+i=pqx + i+pj. – gowers Jul 18 '11 at 22:41
That is right. Nevertheless, in music theory such intervals are usually not understood as actions on a given set (of reals), but are entities themselves. Especially in the mentioned context this group is has more the bookkeeping purpose mentioned in the addition of the OP. In fact the connection to the underlying reals is not unique and seems to be a little bit artificial in the given context. Thus, interpreting “natural” in the sense of music theory, I think this example is an answer to jc's question. – Tobias Schlemmer Aug 22 '11 at 11:37

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