Group theory by geometry

Question

I'm introducing my kids to the concepts of group theory. To make abstract things tangible, I'm trying the geometry way, adopting Arnold's in "Abel's Theorem", so far I've explained, by using symmetry group of triangle, square, and tetrahedron

• group (closure, identity and invertibility, I omitted the associativity part)
• abelian
• subgroup
• cyclic group
• isomorphic

I'm more or less confident as explaining what is a normal subgroup, by interpreting internal automorphisms as "relabeling" -- by algebra style definition internal automorphism is $gag^{-1}$, but one can interpret in a geometry way that $g$ is "relabeling" then $gag^{-1}$ is $a$ under that action.

But then I hesitate on how to move on, for example, how to introduce such concepts in a geometry way, not relying on algebra formula?

• direct product
• coset
• quotient groups
• homomorphism
• commutator & commutant (the commutator subgroup)
• solvable group

How do I introduce such concepts in a geometry way?

Answer 1

Since you are doing this geometrically, I would suggest a different tact on your treatment of normal subgroups: You can think of them as stabilizers of certain extra decorations.

For example, in a dihedral group we could label the various configurations with a right or left handed orientation. Then you obtain the (normal) rotation subgroup as the stabilizer of these orientations.

If the dihedral group is a $D_{2n}$, then you can give two different 2-colorings the vertices and look at the (normal) subgroup which preserves these colorings.

If you look at the group generated by the reflections on the lines here:

https://en.wikipedia.org/wiki/Tetrakis_square_tiling

Then you can give 2-colorings to the triangles. The symmetries are transitive on the triangles, but they don't all preserve colorings. The stabilizer of a coloring is again a normal subgroup.

You can also ask for preservation of the four rotation configurations of the triangles.

There are more complicated things as well:

https://en.wikipedia.org/wiki/Truncated_trihexagonal_tiling#Kisrhombille_tiling

"Wallpaper group" would be a good thing to google.

Quotients by these normal subgroups can be seen as acting on the decorations.

Non-normal subgroups can also be analyzed by what they preserve --- but now they will not be uniform on the undecorated space.

Answer 2

Permit me to amplify @XanderHenderson's comment pointing to Nathan Carter's Visualizing Group Theory, which I think is the perfect vehicle for your intentions.

The MESE question, What makes cosets hard to understand? pointed to his illustration of cosets in $A_4$ over several slides, of which this is one (Download .ppt):

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Here is more on automorphisms, as "rewirings."

Here is a Cayley diagram, showing "interactions between group generators":

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          ^{Image from GroupExplorer.}

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Answer 3

Interesting question, here are some ideas just off the top of my head:

• Direct product groups act on Cartesian products, so the symmetry group of two two squares rotating independently (you could then also talk about how this is a subgroup of the full symmetry group where you're also allowed to swap the squares if they're identical.)

• Cosets: This isn't great, but the cosets of the dihedral group wrt the subgroup of flips are the rotations and the cosets of the rotations wrt are the flips

• Quotient Groups: Same as above, but say "We can ignore the rotations rigorously by forming a new group where we've set the elements of the subgroup of rotations to 1. What will be left in this group?"

• Group homomorphism: Show that the rotation group for the triangle maps into the group for the hexagon (by threefold symmetry), then show that when you try to form a mapping back, you have to explain what happens to the generator $a$. You could demonstrate rotating them together, where you rotate a (marked) hexagon 6 times while rotating a (marked) triangle 6 times and show that rotations respect the groups structure. Then, try it with the pentagon and see that when you've rotated the pentagon 5 times you're not back to the origin for the triangle. A contradiction! You could also use the cube to try this with nonableian groups.

• Look at the commutator subgroup of the dihedral group. First challenge the students to tell you how many transformations of the triangle commute with every other transformation (none). What about the square (one)? Then derive the result.

Someone else is going to have to jump in with solveable groups, it's been too long since I've taken group theory.

Answer 4

There are some references to using the geometric art of M.C. Escher get across group theory (and other) ideas.

For example this article of Marjorie Senechal: https://upcommons.upc.edu/bitstream/handle/2099/1134/st15-07-a3-ocr.pdf

and

https://files.eric.ed.gov/fulltext/ED583835.pdf

More generally you can look for papers related to using the wallpaper groups to teach group theory ideas.