The history of physics, in fact, is a marvel of using simple symmetry principles to construct complicated laws of the universe. Einstein quite famously was able to construct his entire theory of special relativity—the idea that ultimately gave us E=mc2 and explained the heat of the sun—from nothing more than the simple idea that there was no measurable distinction to be made between observers at rest and observers in uniform motion.I do agree that the application of symmetry to physics is the biggest and most pervasive accomplishment of 20th century physics.
The long-overlooked 20th-century mathematician Emmy Noether proved the centrality of symmetry as a physical principle. And what is symmetry—at least as scientists understand it? The mathematician Hermann Weyl gave perhaps the most succinct definition:“A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before.”Which sounds innocuous enough until you realize that if the entire universe were made symmetric, then all of the good features (e.g., you) are decidedly asymmetric lumps that ruin the otherwise perfect beauty of the cosmos.
The seemingly simple idea that the laws of the universe are the same everywhere in space and time turns out to yield justification for long-observed properties of the universe, like Newton’s first law of motion (“An object in motion stays in motion,” etc.) and first law of thermodynamics (the conservation of energy).
As the Nobel laureate Phil Anderson put it:“It is only slightly overstating the case to say that physics is the study of symmetry.”
So who introduced symmetry to physics? Noether's theorem, published in 1918. Hermann Weyl invented gauge theory, and applications of group theory to quantum mechanics.
Einstein did notice in 1905 that the inverse to a Lorentz transformation is a Lorentz transformation. Lorentz credited Einstein for detailing this. This fact would seem to be implicit in Lorentz's explanation of the Michelson-Morley experiment, but Lorentz had not stated or proved it.
The mathematics of symmetry was central to Henri Poincaré's 1905 relativity paper, and that is largely why he deserves the credit for creating special relativity.
The relativity principle and Lorentz group were both named by Poincare. The relativity principle expresses the symmetry between different moving frames of reference, and was defined by Poincare at the 1904 St. Louis Worlds Fair as:
The principle of relativity, according to which the laws of physical phenomena should be the same, whether for an observer fixed, or for an observer carried along in a uniform movement of translation; so that we have not and could not have any means of discerning whether or not we are carried along in such a motion.Einstein took this as one of his postulates in 1905. The Lorentz group is the mathematical expression of those symmetry operations, and Poincare was the first to point out that it was a symmetry group.
These facts are widely acknowledged, even by Einstein scholars. What is not so well known is how much further Poincare took his symmetry analysis. He used the symmetry group to define a non-Euclidean geometry on 4-dimensional spacetime. He proved that Maxwell's equeations for electromagnetism respected those symmetries, and thus could be geometrically realized on spacetime. He looked for a law of gravity that was similarly invariant under the symmetries. For decades afterwards, physicists followed Poincare's example, and used Lorentz invariance as a guiding principle for finding laws of physics.
Einstein has none of this, and showed no sign of even understanding it until hafter Minkowski spelled in out more clearly in 1908. The closest he got was to understand that the Lorentz transformations in one particular direction formed a 1-dimentional group, altho he probably got that from Poincare as Einstein had access to Poincare's first 1905 paper before submitting his own.
Noether, Weyl, and Poincare are three of the most important theoretical physicists of the 20th century. These three are primarily known as mathematicians who only occasionally dabbled in physics, but their theoretical physics was some of the most profound of the time.
This is all detailed in my book, How Einstein Ruined Physics.