I'm today trying to construct a lesson on theory and problems of group theory. A specification of this lesson would be to use as much visual representations as possible. I have found some trivial ways to represent some basic concepts of group theory but I'm stuck with cartesian product.

How can I represent the cartesian product visually?

Thank you for your time.

  • $\begingroup$ I'm in a debate if this would be more of a math.SE question (examples of geometric groups) or relevant here (what is a good way to explain Cartesian products of groups). $\endgroup$
    – Chris C
    Dec 5, 2014 at 22:19
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    $\begingroup$ Simplest example, think of the abelian group $\mathbb{R}, +$. Then the direct product of $\mathbb{R} \times \mathbb{R}$ is just the usual plane with vector addition. Of course, this does not reveal the abstract nature of the group product, but, for visualization it's nice. Next, perhaps the direct product of rotations and additive $\mathbb{R}$. Take a line and attach a circle. The group operation is illustrated by moving the circle along the line and rotating the circle. These motions are independent. $\endgroup$ Dec 6, 2014 at 1:35
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    $\begingroup$ @ChrisC I think this is inherently about something that contributes to understanding a mathematical concept. And since it's in the context of constructing a lesson, I believe there is a strong justification for this as a MESE question. Representation is something that is often considered in Math Education research. $\endgroup$
    – JPBurke
    Dec 6, 2014 at 17:21
  • $\begingroup$ What are the ("trivial") ways you've found to represent basic concepts? It may be possible to extend them to represent the Cartesian product. That would probably be better than introducing a completely new representation. Also, you may find this book helpful: amazon.com/Visual-Classroom-Resource-Materials-Problem/dp/… (in the case of direct product of groups, it uses Caylet $\endgroup$
    – M. Vinay
    Dec 8, 2014 at 3:36

3 Answers 3


Perhaps this YouTube video on Visualizing Group Theory, by Nathan Carter (Bentley Univ), may help. Here is a snapshot illustrating $C_3 \times C_4$:

Blue represents the cyclic group $C_4$ horizontally, and red the cyclic group $C_3$ vertically.

Nathan is the author of the wonderful book,

Carter, Nathan. Visual group theory. MAA, 2009. (MMA link.)

  • $\begingroup$ @Chirac - based on your post I would definitely recommend getting a hold of this book. It has a wealth of visual tools to think about group theory. $\endgroup$ Jan 4, 2015 at 3:04

to illustrate $A \times B,$ draw two orthogonal axis and mark the points of $A$ on one and the points of $B$ on the other. the elements of $A \times B$ are the points where lines through points on the axes meet. or is this too obvious and adequate for the purposes.

  • $\begingroup$ I've only ever understood Group Cartesian products as vectors in the way you describe here. I guess I never really embraced the "Abstract" of "Abstract Algebra" $\endgroup$
    – Richard
    Dec 17, 2014 at 10:46
  • $\begingroup$ This is how I usually think of Cartesian product of sets, but my problem is that it might not help with imagining the group operation. That's the bit that's tricky to imagine for me. $\endgroup$ Dec 17, 2014 at 11:16
  • $\begingroup$ @DavidButlerUofA Doesn't the group operation in a Cartesian product simply decompose just like vector addition (it has been a while since I studies this)? If the components are abelian, then it would be isomorphic to vector addition. If not abelian, I have little intuition of groups, even without the Cartesian product! $\endgroup$
    – Richard
    Dec 17, 2014 at 21:30
  • $\begingroup$ Oh I see. You should say that in your answer! Perhaps even give an example with, say $\mathbb{Z}_3$ and $\mathbb{Z}_4$ to illustrate the group operation part. $\endgroup$ Dec 17, 2014 at 21:57

When I was originally taught groups, one concrete example was the symmetry transformations of an object, like an equilateral triangle ($S_3$) or a rectangle ($V_4$). Usually you label or colour-code the corners so you can tell the difference between them.

One easy way to form a Cartesian product would be to have transformations of both object simultaneously. So an element might be "flip the rectangle horizontally and turn the triangle $120^\circ$". You can imagine the students with an actual rectangle and triangle and doing one action with one hand and one with the other.

A useful thing to do would be to find the order of such an element. So keep doing both simultaneously until both shapes come back to the original configuration. The number of times is the order. They should very easily come up with the idea that the order will be the greatest common multiple of the orders of the two individual transformations.


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