Again, you are calling `research' to the activity of, at best, recognizing symmetry at certain places, and lack of symmetry at others.
To see where can the notion of symmetry take us in this context, you may want to read an introductory text in the theory of finite groups. This topic is more abstract than linear algebra or calculus, and this is why it is usually being taught in the second year of undergraduate studies in mathematics. But in fact it can be done earlier.
For example, I often recommend J.Rotman's book "Introduction to the Theory of Groups". The first four chapters or so are quite easy to read, and contain a lot of interesting material.