Are symmetries necessary in group action?

Question

This could be a very basic question.

Group theory is often described as studying symmetries. I understand that $SO(2)$ could be used to describe a 2D rotationally invariant object (and therefore by studying its symmetries). However, can't it also be used to study rotations of a non-rotationally-invariant object?

I also know that (by Cayley's theorem I think), one can always associate group actions to symmetries of some object.

However, is it absolutely necessary for the group actions to be related to symmetries? I see no reason for this to be the case from the definitions alone.

Would be I correct in saying, all abstract groups can be associated with symmetries of some object (so in that sense symmetries are always present), however, often these objects would end up being quite contrived so really you can study abstract groups without really needing to be thinking about symmetries at all?

Answer 1

I agree.

Symmetry groups are a big application of basic group theory, symmetry is probably the best starting point for motivating groups, but it is not the whole story. Action is more general than symmetry. There are many cases of (isomorphism classes of) groups that either don't, or didn't originally, have an interpretation as a symmetry group that isn't convoluted or artificial or indirect (although I think this will be extremely debatable), or even have multiple different perfectly valid interpretations as symmetry groups.

Possible examples: there are questions about what the alternating groups $A_n$ are the symmetry groups of, and also what 3D figure the quaternion group $Q_8$ might be the symmetry group of. There are probably other examples with many of the exceptional items in the classifications of finite simple groups and simple Lie groups. Exceptional isomorphisms also give ways that groups are symmetry groups in more than one way. Probably the simplest way that comes to mind is $\mathrm{PSL}_2\mathbb{F}_7\cong\mathrm{PGL}_3\mathbb{F}_2$, which acts either as the symmetries of the projective line $\mathbb{F}_7\mathbb{P}^1$ or projective plane $\mathbb{F}_2\mathbb{P}^2$.

That said, I do think actions give purpose to groups. Acting on things is their highest calling. I'd say abstract groups are to group actions as potential energy is to kinetic energy.

Answer 2

First, mathematicians are interested in group actions because they encode symmetries. Why are you interested in group actions, if not for that reason?

Second, every group action has a symmetry associated with it; it's just not a very interesting one. To be precise, let $\kappa$ be a cardinal and $R$ a $\kappa$-ary relation on $X$; that is, $R\subseteq X^{\kappa}$, but we write "$R\{x_j\}_{j\in \kappa}$" rather than "$\{x_j\}_{j\in\kappa}\in R$". Some examples include

• the binary "negation" relation $N$ on the space $\mathscr{L}$ of logical formulas, where ${\phi\mathrel{N}\psi}\iff{\phi=\neg\psi}$,
• the trinary "multiplication" relation $T$ on any group $H$, where ${a\mathrel{T}b\mathrel{T}c}\iff{a=bc}$, and
• the $\omega_0$-ary "limit" relation $L$ on any topological space $Y$, where ${x_{\infty}\mathrel{L}x_0\mathrel{L}x_1\mathrel{L}\cdots}\iff{x_{\infty}=\lim_n{x_n}}$.

These examples are natural symmetries that we should like preserved. To incorporate them, we must allow $\kappa$ to be (relatively) large and we care about the order in which we put the arguments to our relation ($x_{\infty}$ is very different from $x_1$!). (In a moment, I will abuse these requirements.)

For any such relation $R$, let $G$ be the set of bijections preserving $R$, i.e. $$G=\{f:{R\{x_j\}_j\iff R\{f(x_j)\}_j}\}\subseteq X\overset{\sim}{\to} X$$ Then $G$ is a group acting on $X$; in the above examples, $G$ is $C_2^{|\mathscr{L}|}$, $\mathrm{Aut}{(H)}$, and $\mathrm{Homeo}{(Y)}$, respectively.

Conversely, as long as your group action is faithful, $G$ is precisely the group preserving the $|G|$-ary relation $$R_G=\{\{gx\}_{g\in G}:x\in X\}$$ One can then ask interesting questions about minimal $\kappa$ or $R$ to achieve a specific group, but I don't know any good results in this direction.

Answer 3

The definition of $SO(2)$ is the orientation-preserving point symmetries of a Euclidean plane. Then, given some object in the plane, we can try to understand it to see how it changes under this $SO(2)$ action -- that is, understand it by contrasting it to the backdrop of the perfectly symmetric plane.

For example, suppose we have a unit circle $S^1\subseteq \mathbb{R}^2$ centered at the origin and a complex-valued function $f:S^1\to\mathbb{C}$. The $SO(2)$ action of the plane gives an action on such functions by rotating the domain. Maybe this seems rather straightforward, it's just a rotation, but it turns out there is very interesting theory here.

Before that, let's talk about a linear operator $T:\mathbb{R}^n\to\mathbb{R}^n$. Linear operators, if they are invertible, generate groups (the set of all powers of $T$), and they are a kind of point symmetry of $\mathbb{R}^n$. One way to study linear operators (invertible or not) is to look for invariant subspaces. These are subspaces $U\subseteq \mathbb{R}^n$ such that $T(U)\subseteq U$. They give you an idea of sub-symmetries of the action of $T$ on $\mathbb{R}^n$. An interesting fact about $\mathbb{R}$-linear operators is that every invariant subspace always contains a minimal nontrivial invariant subspace of dimension $1$ or $2$. The action of $T$ on these are particularly simple. For a $1$-dimensional minimal invariant subspace, the action is by scaling, and for a $2$-dimensional one, the action is by a combination scaling and rotation (with respect to some basis). In an undergraduate linear algebra course, you meet these in the context of eigenvalues and eigenvectors -- the first case is the span of an eigenvector with a real eigenvalue, and the second case is what happens when you have complex eigenvalues (the polar form giving the scaling and rotation).

Going back to circles, it turns out there are certain special functions that are acted upon in a particular special way, namely $f(\theta)=e^{ni\theta}$ for $n\in\mathbb{Z}$. If you rotate such an $f$ by $\alpha$ radians counterclockwise, then the result is $e^{-ni\alpha}f(\theta)$ -- the function is just scaled. This is analogous to the linear operators digression: we have $SO(2)$ acting on the vector space of all functions $S^1\to \mathbb{C}$, and inside this vector space are $1$-dimensional invariant subspaces for each $n\in\mathbb{Z}$ given by these $\theta\mapsto e^{ni\theta}$ functions. These functions are like eigenvectors in that they're being scaled, but the scaling factor depends on how much you're rotating by. As it turns out, these are all of the minimal nontrivial invariant subspaces for this $SO(2)$ action. Also, by some general theory every nice enough $f:S^1\to\mathbb{C}$ (say square integrable functions) can be written as some infinite series of these $\theta\mapsto e^{ni\theta}$ functions. This is known as the Fourier series.

The idea in general is that if you have a group that can be thought of as the symmetries of some space, then you can decompose things in that space into pieces that change in simpler ways. For the circle, those pieces were frequency components. And for other spaces and other groups many interesting things can happen.