On Group Theory (-1889)

"A set of symbols 1, α, β,..., all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group. [...] These symbols are not in general convertible [commutative] but are associative [...] and it follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor [i.e., on the left or on the right], the effect is simply to reproduce the group." (Arthur Cayley, "On the theory of groups, as depending on the symbolic equation θ^n = 1.", 1854)

"This [...] does not in any wise show that the best or easiest mode of treating the general problem is thus to regard it as a problem of substitutions: and it seems clear that the better course is to consider the general problem in itself, and to deduce from it the theory of groups of substitutions." (Arthur Cayley, 1878)

"A group that is not irreducible [indecomposable] can be decomposed into purely irreducible factors. As a rule, such a decomposition can be accomplished in many ways. However, regardless of the way in which it is carried out, the number of irreducible factors is always the same and the factors in the two decompositions can be so paired off that the corresponding factors have the same order." (Georg Frobenius & Ludwig Stickelberger, "On groups of commuting elements", 1879)

"A system G of h arbitrary elements θ1, θ2,...,θh is called a group of degree h if it satisfies the following conditions: (I) By some rule which is designated as composition or multiplication, from any two elements of the same system one derives a new element of the samesystem. In symbols, θrθs = θt. (II) It is always true that (θrθs)θt = θr(θsθt) = θrθsθt. (III) From θθr = θθs or from θrθ = θsθ it follows that θr = θs." (Weber, 1882)

"The following investigations aim to continue the study of the properties of a group in its abstract formulation. In particular, this will pose the question of the extent to which these properties have an invariant character present in all the different realizations of the group, and the question of what leads to the exact determination of their essential group-theoretic content." (Walther von Dyck, "Group-theoretic studies", 1882)

"Now let there be given a sequence of transformations A, B, C,..., . If this sequence has the property that the composite of any two of its transformations yields a transformation that again belongs to the sequence, then the latter will be called a group of transformations." (Felix Klein, 1880s) 

"The special subject of group theory extends through all of modern mathematics. As an ordering and classifying principle, it intervenes in the most varied domains." (Felix Klein, 1880s)

"Instead of the points of a line, plane, space, or any manifold under investigation, we may use instead any figure contained within the manifold: a group of points, curve, surface, etc. As there is nothing at all determined at the outset about the number of arbitrary parameters upon which these figures should depend, the number of dimensions of the line, plane, space, etc. is likewise arbitrary and depends only on the choice of space element. But so long as we base our geometrical investigation on the same group of transformations, the geometrical content remains unchanged. That is, every theorem resulting from one choice of space element will also be a theorem under any other choice; only the arrangement and correlation of the theorems will be changed. The essential thing is thus the group of transformations; the number of dimensions to be assigned to a manifold is only of secondary importance." (Felix Klein, "A comparative review of recent researches in geometry", Bulletin of the American Mathematoical Society 2(10), 1893)

On Group Theory (2010-2019)

"Combinatorics with all its various aspects is a broad field of Mathematics which has many applications in areas like Topology, Group Theory and even Analysis. A reason for its wide range of applications might be that Combinatorics is rather a way of thinking than a homogeneous theory, and consequently Combinatorics is quite difficult to define. Nevertheless, let us start with a definition of Combinatorics which will be suitable for our purpose: Combinatorics is the branch of Mathematics which studies collections of objects that satisfy certain criteria, and is in particular concerned with deciding how large or how small such collections might be." (Lorenz J Halbeisen,"Combinatorial Set Theory: With a Gentle Introduction to Forcing", 2011)

"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)

"Besides being great history, Galois theory is also great mathematics. This is due primarily to two factors: first, its surprising link between group theory and the roots of polynomials, and second, the elegance of its presentation. Galois theory is often described as one of the most beautiful parts of mathematics." (David A Cox, "Galois Theory" 2nd Ed., [preface] 2012)

"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)

"Often a yes/no problem in mathematics becomes more interesting when it turns into a counting problem. One reason for this is that counting may introduce more structure, or a finer structure, into the data or the conceptual framework of the problem. Similarly, 

sometimes the counting problem becomes more interesting if it can be turned into a group theory problem. Sometimes, the yes/no problem has an obvious answer but the counting problem still yields a fascinating and beautiful theory. Another reason to solve counting problems is that we are more naturally interested in a more precise and more quantitative question than the simple binary puzzle that we began with. Of course, only with success in achieving an interesting answer will we be satisfied with the more complex approach." (Avner Ash & Robert Gross, "Summing It Up : From one plus one to modern number theory", 2016)

"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)

On Group Theory (1900-1949)

"If, then, one takes away from the mathematical theory that which appears merely as an accident, namely its matter, only what is essential will remain, namely its form; and this form, which constitutes so to speak the solid skeleton of the theory, will be the structure of the group." (Henri Poincaré, "Rapport sur les travaux de M. Cartan", Acta Mathematica 38, 1914)

"If we then strip the mathematical theory of what appears in it merely as an accident, that is of its matter, only the essential is left, that is its form; and this form, which constitutes, one might say, the solid skeleton of the theory, will be the structure of the group." (Henri Poincaré, "Rapport sur les travaux de M. Cartan", Acta Mathematica 38, 1914) 

"If indeed one tries to clarify the notion of equality, which is introduced right at the beginning of Geometry, one is led to say that two figures are equal when one can go from one to the other by a specific geometric operation, called a motion. This is only a change of words; but the axiom according to which two figures equal to a third are equal to one another, subjects those operations called motions to a certain law; that is, that an operation which is the result of two successive motions is itself a motion. It is this law that mathematicians express by saying that motions form a group. Elementary Geometry can then be defined by the study of properties of figures which do not change under the operations of the group of motions." (Élie Cartan, "Lec̜ons sur la géométrie des espaces de Riemann", 1928)

"[...] it is the whole logical structure of elementary Geometry which is contained in the group of motions and even, in a more precise manner, in the law according to which operations of that group compose with each other, independently of the nature of the objects on which these operations act. This law constitutes what we call the group structure." (Élie Cartan, "Lec̜ons sur la géométrie des espaces de Riemann", 1928)

"Mathematical research can lend its organisational characteristics to poetry, whereby disjointed metaphors take on a universal sense. Similarly, the axiomatic foundations of group theory can be assimilated into a larger moral concept of a unified universe. Without this, mathematics would be a laborious Barbary." (Dan Barbilian, "The Autobiography of the Scientist", 1940)

"Given any group of transformations in space which includes the principal group as a sub-group, then the invariant theory of this group gives a definite kind of geometry, and every possible geometry can be obtained in this way. Thus each geometry is characterized by its group, which, therefore, assumes the leading place in our considerations." (Felix Klein, "Elementary Mathematics from an Elementary Standpoint: Geometry", 1939)

"The invariant character of a mathematical discipline can be formulated in these terms. Thus, in group theory all the basic constructions can be regarded as the definitions of co- or contravariant functors, so we may formulate the dictum: The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms." (Samuel Eilenberg & Saunders MacLane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms. More precisely, group theory studies functors defined on well specified categories of groups, with values in another such category." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

On Group Theory (1950-1974)

"The notion of an abstract group arises by consideration of the formal properties of one-to-one transformations of a set onto itself. Similarly, the notion of a category is obtained from the formal properties of the class of all transformations y : X → Y of any one set into another, or of continuous transformations of one topological space into another, or of homomorphisms, of one group into another, and so on." (Saunders Mac Lane, "Duality for groups", Bulletin of the American Mathematical Society 56, 1950)

"Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos." (Eric T Bell, "Mathematics, Queen and Servant of Science", 1951)

"The trouble with group theory is that it leaves so much unexplained that one would like to explain. It isolates in a beautiful way those aspects of nature that can be understood in terms of abstract symmetry alone. It does not offer much hope of explaining the messier facts of life, the numerical values of particle lifetimes and interaction strengths - the great bulk of quantitative experimental data that is now waiting for explanation. The process of abstraction seems to have been too drastic, so that many essential and concrete features of the real world have been left out of consideration. Altogether group theory succeeds just because its aims are modest. It does not try to explain everything, and it does not seem likely that it will grow into a complete or comprehensive theory of the physical world." (Freeman J Dyson, "Mathematics in the Physical Sciences", Scientific American Vol. 211 (3), 1964)

"Though determinants and matrices received a great deal of attention in the nineteenth century and thousands of papers were written on these subjects, they do not constitute great innovations in mathematics. [...] Neither determinants nor matrices have influenced deeply the course of mathematics despite their utility as compact expressions and despite the suggestiveness of matrices as concrete groups for the discernment of general theorems of group theory." (Morris Kline, "Mathematical Thought From Ancient to Modern Times", 1972) 

"If indeed one tries to clarify the notion of equality, which is introduced right at the beginning of Geometry, one is led to say that two figures are equal when one can go from one to the other by a specific geometric operation, called a motion. This is only a change of words; but the axiom according to which two figures equal to a third are equal to one another, subjects those operations called motions to a certain law; that is, that an operation which is the result of two successive motions is itself a motion. It is this law that mathematicians express by saying that motions form a group. Elementary Geometry can then be defined by the study of properties of figures which do not change under the operations of the group of motions." (Élie Cartan, "Notice sur les travaux scientifiques", 1974)

"[…] it is the whole logical structure of elementary Geometry which is contained in the group of motions and even, in a more precise manner, in the law according to which operations of that group compose with each other, independently of the nature of the objects on which these operations act. This law constitutes what we call the group structure." (Élie Cartan, "Notice sur les travaux scientifiques", 1974)

On Group Theory (2000-2009)

"By definition, a Kähler manifold is one with a complex structure (this means in particular that the coordinates changes are holomorphic for the complex coordinates) together with a Riemannian metric which has with this complex structure the best possible link, namely that multiplication of tangent vectors by unit complex numbers preserves the metric, but moreover the complex structure is invariant under parallel transport. This is equivalent to the condition that the holonomy group be included in the unitary group, hence equivalent also to ask for the existence of a 2-form of maximal rank and of zero covariant derivative."(Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"Combinatorics could be described as the study of arrangements of objects according to specified rules. We want to know, first, whether a particular arrangement is possible at all, and, if so, in how many different ways it can be done. Algebraic and even probabilistic methods play an increasingly important role in answering these questions. If we have two sets of arrangements with the same cardinality, we might want to construct a natural bijection between them. We might also want to have an algorithm for constructing a particular arrangement or all arrangements, as well as for computing numerical characteristics of them; in particular, we can consider optimization problems related to such arrangements. Finally, we might be interested in an even deeper study, by investigating the structural properties of the arrangements. Methods from areas such as group theory and topology are useful here, by enabling us to study symmetries of the arrangements, as well as topological properties of certain spaces associated with them, which translate into combinatorial properties." (Cristian Lenart, "The Many Faces of Modern Combinatorics", 2003)

"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)

String Theory I

"The odd thing about string theory was very odd indeed. It required that the universe have at least ten dimensions. As we live in a universe of only four dimensions, the theory postulated that the other dimensions [...] had collapsed into structures so tiny that we do not notice them. (Timothy Ferris, "Coming of Age in the Milky Way", 1988)

"Theoretical physicists are accustomed to living in a world which is removed from tangible objects by two levels of abstraction. From tangible atoms we move by one level of abstraction to invisible fields and particles. A second level of abstraction takes us from fields and particles to the symmetry-groups by which fields and particles are related. The superstring theory takes us beyond symmetry-groups to two further levels of abstraction. The third level of abstraction is the interpretation of symmetry-groups in terms of states in ten-dimensional space-time. The fourth level is the world of the superstrings by whose dynamical behavior the states are defined." (Freeman J Dyson, "Infinite in All Directions", 1988)

"No other theory known to science [other than superstring theory] uses such powerful mathematics at such a fundamental level. […] because any unified field theory first must absorb the Riemannian geometry of Einstein’s theory and the Lie groups coming from quantum field theory. […] The new mathematics, which is responsible for the merger of these two theories, is topology, and it is responsible for accomplishing the seemingly impossible task of abolishing the infinities of a quantum theory of gravity." (Michio Kaku, "Hyperspace", 1995)

"Discovery of supersymmetry would be one of the real milestones in physics, made even more exciting by its close links to still more ambitious theoretical ideas. Indeed, supersymmetry is one of the basic requirements of 'string theory', which is the framework in which theoretical physicists have had some success in unifying gravity with the rest of the elementary particle forces. Discovery of supersymmetry would would certainly give string theory an enormous boost." (Edward Witten, [preface to (Gordon Kane, "Supersymmetry: Unveiling the Ultimate Laws of Nature", 2000) 1999)

"For string theory to make sense, the universe should have nine spatial dimensions and one time dimension, for a total of ten dimensions." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

"If string theory is right, the microscopic fabric of our universe is a richly intertwined multidimensional labyrinth within which the strings of the universe endlessly twist and vibrate, rhythmically beating out the laws of the cosmos." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)

"To appreciate string theory, one must understand what it is that is being generalized." (Edward Witten, "Physical Law and the Quest for Mathematical Understanding", Bulletin of the American Mathematical Society Vol. 40 (1), 2002)

"In string theory one studies strings moving in a fixed classical spacetime. […] what we call a background-dependent approach. […] One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves - at the rate predicted by general relativity to the […] accuracy of eleven decimal place - tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up." (Lee Smolin, "Loop Quantum Gravity", The New Humanists: Science at the Edge, 2003)

"String theory is extremely attractive because gravity is forced upon us. All known consistent string theories include gravity, so while gravity is impossible in quantum field theory as we have known it, it is obligatory in string theory." (Edward Witten)

"[The discovery of string theory is like] wandering around the desert and then stumbling on a tiny pebble. But when we examine it carefully, we find that it is the tip of a gigantic pyramid." (Michio Kaku)