Group Theory III

"Group theory is a branch of mathematics that describes the properties of an abstract model of phenomena that depend on symmetry. Despite its abstract tone, group theory provides practical techniques for making quantitative and verifiable predictions about the behavior of atoms, molecules and solids." (Arthur M Lesk, "Introduction to Symmetry and Group Theory for Chemists", 2004) 

"Group theory is a powerful tool for studying the symmetry of a physical system, especially the symmetry of a quantum system. Since the exact solution of the dynamic equation in the quantum theory is generally difficult to obtain, one has to find other methods to analyze the property of the system. Group theory provides an effective method by analyzing symmetry of the system to obtain some precise information of the system verifiable with observations." (Zhong-Qi Ma, Xiao-Yan Gu, "Problems and Solutions in Group Theory for Physicists", 2004)

"Mathematicians have evolved a systematic way of thinking about symmetries that is fairly easy to grasp at the outset and a lot of fun to play with. This almost magical subject is known as group theory. […] Group theory is the mathematical language of symmetry, and it is so important that it seems to play a fundamental role in the very structure of nature. It governs the forces we see and is believed to be the organizing principle underlying all of the dynamics of elementary particles. Indeed, in modem physics the concept of symmetry serves as perhaps the most crucial concept of all. Symmetry principles are now known to dictate the basic laws of physics, to control the structure and dynamics of matter, and to define the fundamental forces in nature. Nature, at its most fundamental level, is defined by symmetry." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"A group is a collection of objects, one that is alive in the sense that some underlying principle of productivity is at work engendering new members from old. […] Like many other highly structured objects, groups have parts, and in particular they may well have subgroups as parts, one group nested within a large group, kangarette to kangaroo." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"But like every profound mathematical idea, the concept of a group reveals something about the nature of the world that lies beyond the mathematician’s symbols. […] There is […] a royal road between group theory and the most fundamental processes in nature. Some groups represent - they are reflections of - continuous rotations, things that whiz around and around smoothly." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Topology, like other branches of pure mathematics such as group theory, is an axiomatic subject. We start with a set of axioms and we use these axioms to prove propositions and theorems. It is extremely important to develop your skill at writing proofs." (Sydney A Morris, "Topology without Tears", 2011)

"The potential freedom in the choice of a particular mathematical representation of physical objects is loosely called symmetry. In mathematical terms, physical symmetries are intimately related to groups in the sense that symmetry transformations form a group." (Teiko Heinosaari & Mario Ziman, "The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement", 2012) 

"Galois and Abel independently discovered the basic idea of symmetry. They were both coming at the problem from the algebra of polynomials, but what they each realized was that underlying the solution of polynomials was a fundamental problem of symmetry. The way that they understood symmetry was in terms of permutation groups. A permutation group is the most fundamental structure of symmetry. […] permutation groups are the master groups of symmetry: every kind of symmetry is encoded in the structure of the permutation group." (Mark C Chu-Carroll, "Good Math: A Geek’s Guide to the Beauty of Numbers, Logic, and Computation", 2013) 

"[…] the role that symmetry plays is not confined to material objects. Symmetries can also refer to theories and, in particular, to quantum theory. For if the laws of physics are to be invariant under changes of reference frames, the set of all such transformations will form a group. Which transformations and which groups depends on the systems under consideration." (William H Klink & Sujeev Wickramasekara, "Relativity, Symmetry and the Structure of Quantum Theory I: Galilean quantum theory", 2015) 

"The theory of groups is considered the language par excellence to study symmetry in science; it provides the mathematical formalism needed to tackle symmetry in a precise way." (Pieter Thyssen & Arnout Ceulemans, "Shattered Symmetry: Group Theory from the Eightfold Way to the Periodic Table", 2017)