Group theory: the study of symmetries?

Question

I understand basic group theory. I would say that I've seen most of the standard stuff up to, say, the quotient group.

I feel like I've seen in more than one place the suggestion that group theory is the study of symmetries, or actions that leave something (approximately) unchanged. Unfortunately I can only find a couple sources. At 0:49 in this 3 Blue 1 Brown video, the narrator says "[Group theory] is all about codifying the idea of symmetry." The whole video seems to be infused with the idea that every group represents the symmetry of something.

In this video about the Langlands Program, the presenter discusses symmetry as a lead-in to groups beginning around 33:00. I don't know if he actually describes group theory as being about the study of symmetry, but the general attitude seems pretty similar to that of the previous video.

This doesn't jive with my intuition very well. I can see perfectly well that part of group theory has to do with symmetries: one only has to consider rotating and flipping a square to see this. But is all of group theory about symmetry? I feel like there must be plenty of groups that have nothing to do with symmetry. Am I wrong?

From my understanding of group theory, this is more along the lines of a way to introduce the topics to people unfamiliar with group theory. It’s kind of like the “topology is the study of shapes that you can mold but not cut” definition of topology. It’s not the only way to think of it, but it’s a good metaphor. — Snacc, Sep 23 '20 at 00:07
every group is isomorphic to a subgroup of the symmetric group — J. W. Tanner, Sep 23 '20 at 00:14
Related MO post: Is every finite group a group of “symmetries”? — Viktor Vaughn, Sep 23 '20 at 00:43

score 1 · Answer 1 · answered Sep 23 '20 at 00:06

1

In view of Cayley's theorem, one could say yes.

Though I don't know that you could say symmetries of a geometrical object. This statement requires some support.

answered Sep 23 '20 at 00:06

3

Every Cayley graph is indeed a geometric object: put a metric on the graph with each edge of length 1. These are the central examples of geometric group theory. – Lee Mosher Sep 23 '20 at 19:34
Thanks for enlightening me @LeeMosher. I have never learned much about Cayley graphs. But probably should. – Sep 23 '20 at 20:16

score 1 · Answer 2 · answered Sep 23 '20 at 00:11

If you have a look at my profile, there's a link to my Master's dissertation on inverse semigroups and inductive groupoids. They generalise the notion of symmetry in group theory to partial symmetries, like instances of self-similarities in certain fractals. In fact, there's a (couple of) nice relationship(s) between inverse semigroups and inductive groupoids that see each of them as an axiomatisation of this wider sense of symmetry, meaning we can move from one to the other.

score 1 · Answer 3 · edited Mar 05 '23 at 15:03

My background is physics not pure mathematics, I am not a general expert on groups, but a great many groups occur in physics and they are associated with symmetries of the Lagrangian. Furthermore, by Noether's theorem, these symmetries are associated with conserved quantities, such as energy, momentum, angular momentum, and charge. In Quantum Field Theory, particles often have "internal" symmetries, that are usually described as a linear group that leaves certain tensors invariant. Most groups that I can think of can be derived as a subset of $GL_n(C)$, or $GL_n(R)$ which leaves specified tensors in the defining representation (and its adjoint and dual) invariant.
In short I would say yes.

score 1 · Answer 4 · answered Sep 23 '20 at 19:30

The question is not necessarily whether or not all of group theory is symmetry, but why it is so natural that groups are connected with various symmetries. Of course, it is natural as well for groups to be connected with solutions of polynomial equations as Galois groups, or with number-theoretic structures used by Gauss and Kronecker, and so on.

A possible extension here is also to consider groups not only as abstract groups, but also as transformation groups, Lie groups, algebraic groups etc. For an interesting post with symmetries and Lie groups see here:

Groups as symmetries, and question about Lie groups

Group theory: the study of symmetries?

4 Answers4