Understanding Group Action: A Guide for Graeme

[FunkyDwarf]

Hey guys

Im not sure if this question belongs in this forum but anyway. Can someone please explain a group action to me? I vaguely understand but the notes i have for it are pretty crap and the interweb hasnt been of much use (neither has my cryptic lecturer). You can take it as a given (obviously) that i know the pre-requisites for groups and how to use them (sort of ) for your answer. Is it simply that for g an element in group G under binary operation * that the 'action' is g*x where x is something else like a point in R2 or something? And that this is special because when you do it for a collection of points (ie an object) it does something significant?

Cheers!
-Graeme

 

[Chris Hillman]

Where/How to Read/Learn About Group Actions

Im not sure if this question belongs in this forum but anyway.

I suggest moving this thread to the modern algebra forum.

Can someone please explain a group action to me? ...You can take it as a given (obviously) that i know the pre-requisites for groups and how to use them (sort of ) for your answer.

Good for you, that is the kind of valuable information we wish more inquirers thought to include!

I assume you are at uni so can obtain math textbooks from your local research library. For a UK undergraduate student the best book might be Neumann, Stoy, and Thompson, Groups and Geometry, Oxford University Press, 1999. An excellent cheap book well worth buying (but a little too concise for a first book) is Cameron, Permutation Groups, in the LMS student text series of Cambridge University Press. A very clear modern algebra textbook which treats group actions very well is Fraleigh, A First Course in Abstract Algebra, 3rd edition.

In all of these, some standout topics to look for are:
(1) role of group actions in group theory itself (Sylow theorems),
(2) application to Polya-Redfield counting theory,
(3) smooth actions by Lie groups, homogeneous spaces, and applications to physics, representation theory, differential equations, etc., etc.,
(4) in particular, Kleinian geometry and its applications to physics, invariant theory, algebraic geometry, differential equations, etc., etc.
(5) the connection between symmetry and information/entropy.

Unfortunately, while the books I mentioned cover the first two topics, and while many books treat the third (huge!) topic, information on the last two topics is currently scattered or even unobtainable except as "folklore" (i.e. from conversations like this rather than from written sources).

For Lie theory and applications to physics, an inspiring but possibly too concise book, also in the LMS student text series, is Carter, Segal, and MacDonald, Lectures on Lie Groups and Lie Algebras. At the graduate level, the best book might be Bump, Lie Groups, Springer.

If you are interested in algebraic geometry, I suggest the readable graduate textbook by Harris, Algebraic Geometry, Springer, which quite properly emphasizes the role of group actions. Indeed, group actions in algebraic geometry gave rise to the earliest work by Klein and Lie c. 1871 on what became Kleinian geometry and Lie theory respectively. What we now call Lie theory (study of Lie groups and Lie algebras) arose as background needed for Lie's Jungentraum of doing for differential equations what Galois had done for polynomials, to use symmetry to decide when a ODE or PDE has solutions which can be found explicitly or implicitly by taking advantage of any symmetries, and if so to find them. This has many applications, for example, it leads to the only known truly general method of attack on nonlinear PDEs! (Unfortunately, it is no magic bullet, but it does often yield interesting solutions with little effort.) It turns that essentially all the methods of solving special types of ODEs which are encountered in elementary courses on ODEs are in fact exploiting symmetries in the manner of Lie. So if you ever wondered whether there is some systematic method of treating DEs, see Cantwell, Introduction to Symmetry Analysis, Cambridge University Press.

For the last topic there are currently no books that I know of or even expository papers, despite the simplicity and fundamental nature of the basic ideas. Fortunately, you can work these out for yourself following the clues in "What is Information Theory?" https://www.physicsforums.com/showthread.php?t=183900 once you know about stabilizer subgroups and lattices (in the sense of lattice theory). A book I found helpful for the latter is Priestly and Davis, Introduction to Lattices and Order. After you've read parts of the textbook by Harris, I suggest the natural action by the symmetric group $S_n$ on an n-set and the natural action by the projective group $PGL(n+1,C)$ on $CP^n$ and its analogs over Galois fields of characteristic greater than two as "canonical exemplars" illustrating the basic ideas. Also, for the Lorentz group try a Wikipedia article I wrote (in the last version I edited--- look for "Lorentz Group" at http://en.wikipedia.org/wiki/User:Hillman/Archive. I have also posted some more detailed expositions in various InterNet forums on some of this stuff at various times from 1994-present, which might also be helpful.)

Is it simply that for g an element in group G under binary operation * that the 'action' is g*x where x is something else like a point in R2 or something? And that this is special because when you do it for a collection of points (ie an object) it does something significant?

The formal definition of group action basically says that a left action by G is a group homomorphism to a group of transformations on some set, where it is standard to compose transformations from right to left. If we compose from left to right we get a right action. Watch out! Many authors fail to state that they are dealing with left or right actions or what order they are using to compose transformations, which can cause confusion when comparing textbooks! Anyway, from this you can figure out an alternative formal definition which "hides" what is going on by saying that a left action by the group G on the set X is defined by an operation $G \times X \rightarrow X$, written $x \mapsto g .x$ and satisfying certain laws, in particular $e.x = x$ for all $x \in X$ and $g_2 \, . \, ( g_1 . x) = \left( g_2 \, g_1 \right) \, . \, x$, where you can see that we are "secretly" composing transformations from right to left. That is, if our homomorphism is $\theta:G \rightarrow {\rm Sym}(X)$, then $\theta(g_2) \circ \theta(g_1)$ means perform $\theta(g_1)$ first and $\theta(g_2)$ second. Then $\theta(g_2) \circ \theta(g_1) = \theta(g_2 \, g_1)$ which explains what is going on "under the hood" when I asserted that $g_2 \, . \, ( g_1 . x) = \left( g_2 \, g_1 \right) \, . \, x$.

Another key idea which can be very helpful is that for a given group G, the sets X equipped with a (left/right) action by G form a category, the "category of (left/right) G-sets", which in many ways is somewhat analogous the category of R-modules for a fixed ring R. When I was an undergraduate I used this as a guide to suggest systematically working about many elementary facts about G-sets and I learned a lot that way! Note that the appropriate notion of morphism, a (left) G-hom $\varphi:X \rightarrow Y$ satisfies $\varphi(g.x) = g.\varphi(x)$, and is also called an equivariant map.

I should probably have mentioned, as another standout topic, the application of transformation groups to the theory of measurable transformations and applications to ergodic theory. If you are interested in dynamical systems, you should see Pollicott and Yuri, Dynamical Systems and Ergodic Theory, another book in the LMS student text series of Cambridge University Press. Here, generalizing from the transformation group generated by one transformation (the analog of a cyclic group generated by one element) to abelian transformation groups generated by several commuting transformations has dominated research in symbolic dynamics at the end of the last century. If you are interested in number theory, several recent posts by Terry Tao at http://terrytao.wordpress.com/ discuss connections between abelian group actions, ergodic theory and additive number theory. Be sure to read about the most important theorem in mathematics, the Szemeredi lemma, which can be profitably viewed as the central result in ergodic Ramsey theory.

If you have or develop a serious interest in combinatorics or category theory, you should check out the theory of structors, aka combinatorial species, which perfectly captures the notion of an combinatorial structure placed on a "naked n-set" and which leads to a theory of enumeration vastly generalizing Polya-Redfield theory, in fact basically re-expressing much of the elementary theory of enumerative combinatorics. I found it valuable to rewrite Wilf Generatingfunctionology in terms of this theory! Here too we encounter beautiful connections between group actions and first order logic. To begin reading about these topics, try http://www.arxiv.org/abs/math/0004133

If you are interested in algebraic topology, you will certainly want to learn about the theory of covering spaces, where group actions play a crucial role. See Hatcher, Algebraic Topology. Note that a generalization of Cayley graphs, called Schreier graphs, can be used to define actions by a finitely generated group G on a finite set X, and this has beautiful connections with covering spaces and with the theory of presentations of groups! See the high school(!) textbook by Magnus and Grossman, Groups and their Graphs, Mathematical Association of America, and note that here you definitely want to use right actions, not left actions, in order to read the Schreier graphs consistently. An appendix to another readable textbook, Massey, Algebraic Topology (which only covers homotopy theory, not homology theory), is actually a very good brief survey of the first three pillars of the elementary theory of G-sets.

I should probably mention that both right and left actions are important and occur together in some crucial places, e.g. the theory of double coset spaces which gives the orbital decomposition theorem, the fourth pillar of the elementary theory of G-sets. In some recent issues of This Week in Mathematical Physics, John Baez has discussed (among many other things) a groupoid approach to double cosets, inspired by his reading of Klein's famous 1871 essay, known as the Erlangen Program, which in the late nineteenth century was an organizing principle similar in importance to category theory itself in the late twentieth century. His work may suggest directions in which some of these ideas may develop in the current century. When he speaks of "invariant equivalence relations", note that these are a key topic in model theory (see Fraisse theory in the textbook by Cameron already mentioned).

 

[FunkyDwarf]

Wow thanks that was very...comprehensive. I don't know that much about homomorphisims but I am looking that up at the moment. So was i more or less right in my sloppy definition of a group action? An element of X acted on (from left or right as you mentioned) by an element of the group using the group operation.

My interest is more in physics at the moment and this course is a requirement for it (second year) but i do have an interest in certain maths areas; calc, number theory and vector space stuff which i no realize is useful for Fourier theory and whatnot. I am not sure if i can get those books here but ill have a look.

Thanks a lot for the advice!
-G

 

[Kummer]

Given a group G and a non-empty set X a "group action" is a binary operation $$G\times X\mapsto X$$ so that $$ex=x$$ and $$g_1(g_2 x) = (g_1g_2)x$$. That is all. Group actions are of interest when we let X be the set of all subgroups of G and using certain facts about group actions we can get some interesting and important results.

 

[FunkyDwarf]

when you say ex though you mean acting e on x using the binary operation of the group though right?

 

[matt grime]

when you say ex though you mean acting e on x using the binary operation of the group though right?

No, he *can't* mean that. Since x is not in G, it is just some set, then one cannot compose e with x using the group law of G.


Think of some concrete examples, that always helps. G the set of nxn invertible matrices over R acts on R^n. The group of permutations of {1,..,n} acts on {1,..,n} by, well, permuting things.

 

[FunkyDwarf]

Then what does ex represent? what operation is occurring? it can't just be normal multiplication because that wouldn't be abstract enough

 

[matt grime]

It depends on the situation. Look at the examples. In one case it is normal matrix multiplication. In the other you are applying a function.

In fact the operation can always be thought of as applying a function, if you really have to think of it that way. An action of a group on a set X is a homomorphism from X into the set of all bijective functions from X to itself. This however is not particulary helpful - you never think of matrices as acting as functions on vector spaces, do you?

ex is the element that e maps x to. That is always x itself. gx is the element g maps x to. What that is, or the smartest way to describe it depends on the situation.

 

[Chris Hillman]

Some Clarification

Kummer sketched the defination of a left action; adding a bit of detail we obtain a formal definition:

Given a group G and a non-empty set X, a left action by G on X is a binary operation defined by a map $$\theta:G\times X \rightarrow X$$ satisfying the axioms

• $e . x = x$, where e is the identity element of G and where x is any point in X,
• $g_2. (g_1 . x) = (g_2 \, g_1) . x$, where $g_2, \, g_1$ are any elements of G and x is any point in X.

Note that here I have written $\theta(g,x) = g.x$, a fairly standard notation which is intended to prevent confusion of the two binary operations: the group product (denoted by juxtaposition) and the action (denoted by a period). For example, in the law
$$g_2 . \left( g_1 . x \right) = \left( g_2 \, g_2 \right) . x, \; {\rm for} \; {\rm all} \; g_1, \, g_2 \in G, \; x \in X$$
on the right hand side, $g_2 \, g_1 = g_3$ is the group product, while $\left( g_2 \, g_2 \right) . x = g_3 . x$ is the element obtained by moving the point $x$ via $g_3$ according to the action defined by $\theta$.

This comment addresses your question about the meaning of $e. x = \theta(e,x)$: if we are given a binary operation as a map $\theta:G \times X \rightarrow X$ which satisfies the above axioms, in particular has the property $\theta(e, x) = x$ for all x in X, then this operation defines a left action by G on X. You can check for example that we obtain a left action by G on the set of subgroups of G by conjugation, $g.H = g \, H \, g^{-1}$. (If we use $H.g = g^{-1} \, H \, g$ we obtain the dual right action. More generally, given a left action written $x \mapsto g.x$, we can define a dual right action by $x \mapsto x.g = g^{-1}. x$, where the group theoretic identity $\left( g_2 \, g_1 \right)^{-1} = g_1^{-1} \, g_2^{-1}$ is crucial to ensuring that the axiom $\left( x.g_1 \right) . g_2 = x . \left( g_1 \, g_2 \right)$ is satisfied.)

Another convenient and suggestive notation is $\theta (g,x) = \theta_g (x)$, which stresses that given an action by G on X, each group element g "induces" a transformation of X, i.e. a bijection $\theta_g: X \rightarrow X$, defined by $\theta_g(x) = g.x$. Then the transformations $\theta_g, \; g \in G$ form a group, which is a factor group of G. This provides the connection between the "abstract" notions of the category of left G-sets (sets equipped with a left action by G) and the theory of transformation groups and permutation groups. It might be helpful to consistently speak of "elements" of the group G and "points" of the set X, and to keep in mind that elements move the points within X; this should help you to keep straight the different roles played by G and X.

(Historically, the theory of permutation groups arose first, mainly in the work of Lagrange and his contemporaries, then more general transformation groups, in the work of Klein and his contemporaries, then group actions were defined in the modern style near the turn of the last century. This was a slow process because the modern notion of a group did not become fully articulated until late in the nineteenth century!)

When Kummer added

Group actions are of interest when we let X be the set of all subgroups of G and using certain facts about group actions we can get some interesting and important results.

this could be read as claiming that the only actions which are of interest are various actions by G on certain sets constructed from G itself, such as

• the left action by G on G defined by $g.x = g x$ (the left action by "translation" constructed using the group product in G),
• the left action by G on the set of subgroups of G which is defined by $g.H = g \, H \, g^{-1}$ (the left action by "conjugation").

This is not true, as you can see from my post--- the notion of group action is extremely versatile, arises virtually everywhere in mathematics, and is often very useful for many purposes. I assume that Kummer meant only that actions by G on itself and on its own subgroups constitute the most important type of action for you to pay attention to right now, since you are learning about group theory and these actions are among those involved in Wielandt's elegant proof of the Sylow theorems via suitably chosen actions. If so, I probably agree. The one book I would probably recommend for you right now would be the very readable undergraduate textbook by Neuman, Stoy, and Thompson, Groups and Geometry which I already cited.

In any case, my intent was not to overwhelm you but to give you a glimpse of the smorgasbord which awaits once you learn some basic ideas. No-one expects you to go right out and start reading all the books I recommended! Or even to ever read them all. Rather, I provided an extremely sketchy survey intended to guide your future reading (next summer?); the intent is that you can pick and choose whatever seems interesting to you when you have some free time for extracurricular mathematical study.

 

[morphism]

I think one of the most useful things to note in regards to group actions is that an action of G on X induces a homomorphism of G into Sym(X), the symmetric group on X. This little fact has far-reaching implications, and has been mentioned a couple of times already. I just felt the need to stress it.

 

[Chris Hillman]

I think one of the most useful things to note in regards to group actions is that an action of G on X induces a homomorphism of G into Sym(X), the symmetric group on X. This little fact has far-reaching implications, and has been mentioned a couple of times already. I just felt the need to stress it.

Yes indeed; this is what I was talking about when I said that the transformations $\theta_g:X \rightarrow X, \; g \in G$ form a group. This group is a subgroup of Sym(X) and also a factor group of G. The latter follows from the existence of a group homomorphism $g \mapsto \theta_g$. In fact, as I mentioned in my Post #1, a left action by G on X can be alternatively defined as a homomorphism $\Theta: G \rightarrow \operatorname{Sym}(X)$, where it is understood that we compose the transformations in Sym(X) from right to left. Needless to say, here $\Theta(g) = \theta_g = \theta(g, \cdot)$.

Those of you who know about (linear) representations of groups will recognize that we are saying that the theory of G-sets is the theory of representations of G by transformation groups (if these are finite, they are usually just called permutation groups). So the group ring is important in invariant theory, in representation theory, and in the theory of group actions.

I've been avoiding talking about stabilizers and the dual lattices of (pointwise) stabilizers $\triangleleft A \leq G$ and fixsets $\triangleright \triangleleft A \subset X$, but these are the most important concepts not yet mentioned. (The arrow notation has the same meaning as in the book by Priestly and Davies which I cited earlier.) A good exercise for Graeme would be to try to figure out the effect of a group homomorphism $\psi: G \rightarrow H$ on the lattice of subgroups! The answer is memorable and easy to prove from standard group theory, and it is highly relevant to understanding the simple relationship between symmetry and information which I have discussed in some other posts in various places. The intersection of the stabilizers, sometimes called the "radical" by a formal analogy with a similar situation in the theory of certain algebras, is a normal subgroup of G. In fact, it is $\ker \Theta$ and the quotient group $G/\ker \Theta$ is isomorphic to $\Theta(G) \leq \operatorname{Sym}(X)$. Graeme, you probably know why that is true! It follows that the subgroup lattice of $\Theta(G)$ and $G$ are simply related. In fact, if $\Theta$ is one-one, they are isomorphic as lattices and we have a faithful action (analogous to faithful representation in the theory of linear representations).

By the way, should there be any lurkers hereabouts with an interest in category theory, the category of G-sets has the remarkable property that it forms an elementary topos. That has many powerful implications, for example, "spaces" of G-homs are themselves automatically objects in the same category. We have exponential objects $X^Y$ where $X, \; Y$ are G-sets and we have a classifying object associated with a notion of logic. Roughly speaking, this means that we could use G-sets (for our favorite group) rather than sets as the foundation for mathematics. (Before you ask, AFAIK it wouldn't make much difference in daily life. There are however other topoi in which "everything is constructible", and so on.)

 

[FunkyDwarf]

if we are given a binary operation as a map $\theta:G \times X \rightarrow X$ which satisfies the above axioms

But this binary operation is not the group operation right? Sorry if I am missing something obvious here, can anyone provide an explicit concrete example so i can see what the hell is going on :P I don't understand when you say an element of g acting on x, i don't understand what operation you are performing.

EDIT: Ok after reading some stuff i think i sort of understand. Youre saying the action is just some arbitary function you define such that certain conditions (already mentioned) hold and that it sends elements of X back into X and that sometimes this is interesting and useful in certain circumstances, that about right? What i still don't understand is what i actually do when i get an element of G and 'act' it on an element of X, I am not sure what operation i should be performing.

 

[morphism]

It's a function you define. For example let G be any group and X be any set. Here's an action of G on X: g*x=x for all g in G and x in X.

Let's check that it satisfies the definition of a left group action:
1*x=x -- check;
(g*h)x = x = g*(h*x) -- check.

It's also obviously a right action.

For another easy example, take S_n and let it act on X={1,2,...,n} in the obvious way, namely let each element of S_n permute the elements of X. So for instance, if we take (1 n) in S_n, then (1 n)*1=n, (1 n)*2=2, ..., (1 n)*(n-1)=n-1, (1 n)*n=1. Try proving that this is a bona fide group action.

Yet another example: take any group G and let it act on itself (so X=G in this case) by left multiplication (g*h = gh) or by conjugation (g*h = g^-1 h g). Both these define actions (the first one is not always a right action though; can you see when it is?). We can also replace X by the set of left (or right) cosets of G, the set of subgroups of G, a fixed subgroup H, etc.

If it's still not clear, google for more examples -- I'm sure you'll find a ton. If it's still not clear then, either go speak with your lecturer or grab one of the books Chris suggested (Dummit & Foote also has nice sections on group actions).

 

[FunkyDwarf]

so youre saying we define * ( in your example)

 

[matt grime]

can anyone provide an explicit concrete example so i can see what the hell is going on

Yes. I can. I did. I gave you two examples - GL(R,n) acting on R^n, or any other matrix group. Any permutation group acting on the set it permutes

What i still don't understand is what i actually do when i get an element of G and 'act' it on an element of X, I am not sure what operation i should be performing.

You have a homomorphism from G to Aut(X), the set of bijections from X to itself. They are then just functions. You apply the function to the element. If G is S_n, g=(12), then g.1=2, g.2=1, and g.x=x for any x not equal to 1 or 2.

 

[FunkyDwarf]

Ok i don't know the groups you mention by name or abbreviation but i ithink i understand now anyway, sort of.

Thanks guys