# Groups as symmetries of objects

## Main Question or Discussion Point

So it's said that every group is a symmetry group of some tangible object. For example, ##S_3## is the symmetry group of ##\{1,2,3 \}##, and ##D_{2n}## is the symmetry group of an n-gon. But what is ##GL_{10} (\mathbb{R})## the symmetry group of? What about ##\mathbb{Z}##?

I have found two theorems that get at this I think: Cayley's theorem and Frucht's theorem. The former says that every group G is isomorphic to a subgroup of the symmetric group acting on G. The latter says that every finite group is the group of symmetries of a finite undirected graph.

Which one gets at the heart of what I'm asking, and also, what would one recommend to better understand groups as symmetry? My textbook takes the standard approach with defining a a group algebraically and then moving on to results, without the motivation regarding symmetry.

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Stephen Tashi
what would one recommend to better understand groups as symmetry?
First I recommend distinguishing among 3 concepts:
1. A group
2. A group action
3. A group representation

So it's said that every group is a symmetry group of some tangible object.
Examples where a group is a "symmetry group of some tangible object" are examples of group actions.

How shall we define an action that is a "symmetry" ? What is a "symmetry". Taking a very general view, let ##S## be a set and let ##F(x)## be some function from ##S## to some other set ##W## (for example ##W## might be ##\mathbb{R}##). Let ##g## be a 1-to-1 function that maps ##S## onto itself. We can define "##g## is a symmetry of ##S## with respect to ##F##" to mean that ##F(s) = F(g(s))## for each ##s \in S##.

Is that a good definition? Can it be made to apply to geometric examples where the idea of a symmetry is a mapping that "brings an object into coincidence with itself"?

I have found two theorems that get at this I think: Cayley's theorem and Frucht's theorem. The former says that every group G is isomorphic to a subgroup of the symmetric group acting on G. The latter says that every finite group is the group of symmetries of a finite undirected graph
It's hard to do better than that!

But what is ##GL_{10} (\mathbb{R})## the symmetry group of?
Cayley's theorem applies.
Or you could invent a function ##F## of 10 variables that is constant under 1-to-1 linear transformations of those 10 variables.

By the way, the notion of ##GL_{10}(\mathbb{R})## is an example of a group representation.

Cayley's theorem also applies.

.

fresh_42
Mentor
By the way, the notion of ##GL_{10}(\mathbb{R})## is an example of a group representation.
This is not quite correct as stated. ##G \longrightarrow GL_{10}(\mathbb{R})## would be an example of a group representation. ##GL_{10}(\mathbb{R})## alone is simply a matrix group. What should it be the representation of? Itself? No action and no representation is needed, only the matrix multiplication. That matrices can be interpreted as linear transformations is another subject. It is also important that ##GL_{10}(\mathbb{R})## is a Lie group, not a representation of one.

fresh_42
Mentor
But what is ##GL_{10} (\mathbb{R})## the symmetry group of? What about ##\mathbb{Z}?##
##GL_{10}(\mathbb{R})## is the symmetry group of ##\mathbb{R}^{10}## and ##\mathbb{Z}## of itself.

This might be disappointing at first sight. However, you have chosen infinite groups, so you cannot expect them to leave only finite objects as the vertices of an n-gon or a finite set of numbers invariant; something infinite is needed. So ##\mathbb{Z}## as translations leaves the one dimensional lattice ##\mathbb{Z}## invariant, and ##GL_{10}(\mathbb{R})## the points in a ten dimensional Euclidean, real vector space. ##SL_{10}(\mathbb{R})## would leave concentric ten dimensional circles (nine spheres) invariant. But as you allowed stretches it's the entire space.

Correction, kudos to @Infrared :
Infrared said:
You said in the thread "Groups as symmetries of object" thread that "$SL_{10}(\mathbb{R})$ would leave concentric ten dimensional circles (nine spheres) invariant". This is false even if you're only talking about spheres centered about the origin, as $\begin{pmatrix} 2 & 0\\0 & 1/2\end{pmatrix}\in SL_2(\mathbb{R})$ takes $(1,0)$ to $(2,0)$. Maybe you confused $SL_n$ and $O(n)$?
Yes, thank you for pointing this out.

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fresh_42
Mentor
Which one gets at the heart of what I'm asking, and also, what would one recommend to better understand groups as symmetry?
This depends on your goals. Cayley is important as it has relatives such as Lie groups are matrix groups, algebras are quotients of tensor algebras, and similar.
Usually in the context of group theory symmetries refer to geometric objects and usually finite groups. If you study physics, then Noether's theorem becomes very important, and this deals with symmetries of differential equation systems and usually infinite matrix groups.
My textbook takes the standard approach with defining a a group algebraically and then moving on to results, without the motivation regarding symmetry.
... which are likely finite groups, or some continuous rotation groups as the orthogonal groups. This plays a role in e.g. crystallography and as you mentioned graph theory. As soon as you are asking for symmetries and invariances, you let a group act on something, which by the way is equivalent to a representation of this group, so both concepts are actually the same. This is an important concept as it is often more important what a group does to an object, than it is how the group itself is structured. And often one provides insights for the other, e.g. remember the orbit formula. Furthermore, a group always acts on itself, per conjugation or left multiplication.

Stephen Tashi
Several questions of terminology have come up.

Some authors use the phrase "representation of a group" to mean what other authors call a "linear representation of a group". Those that distinguish between a "representation" and a "linear representation" presumably agree that a "representation" of a group and a "group action" are the same mathematical structure.

If we are to discuss "symmetries" and "invariances", are we talking about two different things? Do those terms have formal definitions or are they informal descriptions?

Obviously one can use such words as part of a phrase such as "a symmetry of a square" or "an invariant function" and define the phrase in a narrow context. But is there a definition for the general concept of "a symmetry" and "an invariance"? I can compose definitions that satisfy myself, but I haven't see such definitions in textbooks.

Textbooks can even be vague about particular phrases. For example, if a "symmetry of a polygon" is defined to be a "transformation that brings the polygon into coincidence with itself", what does it mean for the polygon to be in "coincidence with itself"? I can formulate a definition for "in coincidence with itself", but I've never seen a textbook that bothered to do this.

My textbook takes the standard approach with defining a a group algebraically and then moving on to results, without the motivation regarding symmetry.
Perhaps "symmetry" and "invariance" are informal terms. For example, perhaps "invariance" is terminology that suggests a lot of specific phrases that include the word "invariant" like "invariant function", "invariant differential equation", "invariant vector" etc. If "symmetry" and "invariance" are informal terms then you can't expect a treatment of group theory to relate them to the general idea of groups except by presenting a lot of specific examples.

fresh_42
Mentor
Some authors use the phrase "representation of a group" to mean what other authors call a "linear representation of a group". Those that distinguish between a "representation" and a "linear representation" presumably agree that a "representation" of a group and a "group action" are the same mathematical structure.
##G \longrightarrow \operatorname{Aut}(X)## is a representation.
##G \longrightarrow GL_n(\mathbb{F})## is a linear representation.
Both are equivalent to ##G## operates (= acts) on ##X##, resp. ##\mathbb{F}^n##.

I have never ever seen it differently. In neither case are the groups ##\operatorname{Aut}(X)## or ##GL_n(\mathbb{F})## alone called a representation, as they are groups in their own rights. The homomorphism is the representation.

Stephen Tashi
I have never ever seen it differently.
For example, from the Wikipedia:
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.
The article continues and gives the more general definition of representation as alternative terminology.

In neither case are the groups ##\operatorname{Aut}(X)## or ##GL_n(\mathbb{F})## alone called a representation, as they are groups in their own rights. The homomorphism is the representation.
I agree. ##GL_n(\mathbb{F})## the image of a group under the homomorphism. By analogy, we should not say "##x^2## is a function". Instead ##x^2## is the image of ##x## under a function.

fresh_42
Mentor
Stephen Tashi
This is because Wikipedia is very biased towards physics
Musn't complain about a bias toward physis on physicsforums.com.

Wikipedia is no reliable source.

If you have the Dover book "Group Theory" by W.R. Scott, look at how he defines a group representation and a "linear group". (I'm not at my house now. I can quote it when I get home.)

You'll find the definition in "Group Representations in Probability And Statistics" by Persi Diaconis even more repulsive.

From "Basic Algebra II: Second Edition" by Nathan Jacobson page 247, Definition 5.1
By a representation ##\rho## of a group ##G## we shall mean a homomorphism of the group ##G## into the group ##GL(V)## of bijective linear transfomations of a finite dimensional vector space ##V## over a field ##F##

fresh_42
Mentor
All this does not change the fact, that these are examples, not definitions. A representation is not automatically linear, e.g. on a symmetry group it is not. I don't complain about Wikipedia, I know how to read it, and that means: do not take it for granted. Wikipedia is not a reliable source for PF. If authors restrict themselves, then they probably won't need other examples. Me, too, has books which only deal with linear representations. So what? However, if authors sell their restricted versions as general definition, then it is simply wrong. All you have is ##e.x=x## and ##g.(h.x)=(gh).x## Anything else is an application for certain situations. Repeating the wrong does not make it right.

Stephen Tashi
However, if authors sell their restricted versions as general definition, then it is simply wrong. .
From the viewpoint of logic, definitions are arbitrary stipulations. Their rightness or wrongness is a matter of cultural norms. There are a significant number of respectable mathematicans who define a group representation in the manner of Jacobson. Jacobson presents a definition of a group representation. He doesn't present his definition as an example. When it comes to cultural norms there can be competing cultural norms. Each side can claim the other side is "wrong".

fresh_42
Mentor
From the viewpoint of logic, definitions are arbitrary stipulations. Their rightness or wrongness is a matter of cultural norms. There are a significant number of respectable mathematicans who define a group representation in the manner of Jacobson. Jacobson presents a definition of a group representation. He doesn't present his definition as an example. When it comes to cultural norms there can be competing cultural norms. Each side can claim the other side is "wrong".
It is not a cultural norm, that a finite group acts as a permutation group of a finite set, it is not a cultural norm that groups in a principal bundle act on fibers, it is not a cultural norm that a group acts via conjugation or left multiplication on itself - none of which is linear at prior.

A linear representation of a group is an example for a representation, not it's definition.
It might be the definition of representations he investigates in his book, but it is not true, that this is the general case. If he claims it, then Jacobson is wrong. Full stop.

By a representation ##\rho## of a group ##G## we shall mean a homomorphism of the group ##G## into the group ##GL(V)## of bijective linear transfomations of a finite dimensional vector space ##V## over a field ##F##

Stephen Tashi
I'm merely citing authorities.

fresh_42
Mentor
I'm merely citing authorities.
But you don't quote them correctly, as I've shown in your Jacobson quotation. He is speaking of only linear representations, so in order not to write linear a thousand times, he says he will omit it, when he said that he will interpret a representation as linear. The (wrong) conclusion that all representations are linear has been yours, not his.

Stephen Tashi
But you don't quote them correctly, as I've shown in your Jacobson quotation.
Are you referring to putting the phase "we shall mean" in bold type? The phrase "we shall mean" is very appropriate to a definition.

He is speaking of only linear representations, so in order not to write linear a thousand times, he says he will omit it, when he said that he will interpret a representation as linear.
Sorry, I didn't see where he made those statements. What page are they on?

fresh_42
Mentor
Are you referring to putting the phase "we shall mean" in bold type?
Yes.
The phrase "we shall mean" is very appropriate to a definition.
No it isn't. It describes a convention made for certain reason. A definition uses the verb to be.
Sorry, I didn't see where he made those statements. What page are they on?
He said, we mean by, which indicates a special situation, a so called convention. In this case for the sake of simplicity and not to be forced to repeat linear over and over again.

Now one and for all:

A presentation of a group ##G## is a pair consisting of a set ##X## and a group homomorphims ##\rho\, : \,G \longrightarrow \operatorname{Aut}(X)##. We can equivalently say that ##G## operates or acts on ##X## by setting ##\rho(g)(x) =: g.x## and we require ##e.x=x## and ##g.(h.x)=(gh).x##

A representation is not necessarily linear, which has been shown by the examples in posts #7, #9, #15.

And once again: repeating the wrong, no matter how often, does not make it right. Thread closed.

• weirdoguy