Over the past few years, I've been doing a lot of self education in the Quantum Mechanics and General Relativity, and of course, there are mathematical elements of both doctrines that are matrices. More specifically, linear operators and transformations. Most of my texts introduced these ideas as fundamental, and not part of some larger mathematical structure--Group Theory.
One thing throughout quantum mechanics that I've noticed is that there is a particular interest in matrices $ A $, such that $Ax = \lambda x$. The Eigenvalue Equation. This linear operator can be said to exhibit symmetry, since it leaves the operand the same up to a constant. For $ \lambda = 1 $, the transformation does indeed leave the operand the same. After some digging, this particular element of quantum mechanics that represents physical measurement, is an element of the Symmetry Group ( call it $ SG(n) $ where $n$ is the dimensionality).
Another example is for the Lorentz transformations in special relativity. Each transformation leaves the reference frame the same no matter where you are. I've usually seen it explained that the Lorentz transformation is just the result of a rotation, very similar to that of the Rotation Group $ O(3) $, but since we have a time dimension and we rotate spacetime and suppress the two other spatial variables, we get the Lorentz Group: $ SO(3,1) $, equipped with 3 space + 1 time, dimensions. Both of these groups have been shown to me to be elements of $ SG(n)$ as well.
At present it seems that Relativity and Quantum Mechanics are the two pillars that our ultimate goal of Cosmology rests upon. From the lens of Group Theory and by the relation of the elements in the theory aforementioned, I am compelled to conclude that the physical laws and principles that are being discovered present day forward are mathematical elements in this group theory that have meaningful physical and mathematical interpretation down at the physical doctrine level, ie the Energy Operator and the Hamiltonian which brought us to the Schrodinger equation, and the Lorentz Transformation that defined special relativity. This also compels me to believe that further advancement can be found if we have our mathematicians race to discover more and more elements in this larger Group Theory, and have Physicist determine and interpret any Physical meaning in the discovered constructs.
My question is then a question of how valid my conclusion is, and how sound does this in turn make my belief.