Sets And Groups

Type of physical science: Mathematical methods

Field of study: Geometry

Sets and groups are fundamental mathematical structures used in the formal representation and analysis of collections of objects, their properties, operations, and their symmetries. Though originally in the domain of pure and abstract mathematics, they have become standard tools in chemistry, physics, and other sciences.

Overview

A set is a collection of objects called elements. As such, it is among the most primitive and essential concepts in reasoning. Examples of sets are the set of all books on a given shelf or the set of all particles in an cloud chamber. Note in the two examples that a set is generally specified within a certain domain of discourse or a universal set. In this way, a set represents a discussed property, distinguishing objects that are in the set (called the characteristic set) and those that are not (called the complement).

A set is termed "finite" if the number of elements in it is finite; otherwise, a set is termed "infinite." Examples of infinite sets include the set of integers, the set of points on a plane, and the set of functions F (x) of a real variable x.

Given a set S, a binary operation on S can be defined as a function *, which takes two elements of S (for example, a and b) as input and produces another element of S (say c) as output. This is commonly denoted as a * b = c. In essence, an operation "combines" or replaces two elements in S by a new element in S. A group is an important type of binary operation on a set.

A binary operation * on set G is called a group if the following three conditions hold (simultaneously): First, for every three elements of a, b, c in G, the operation * is associative, namely that (a * b) * c = a * (b* c); second, there exists an element e in G (called the identity) for which a * e = e * a = a for every element a in G; third, for each element a in G there exists a corresponding unique element called the inverse of a and denoted as a^-1, such that a * a-1 = a^-1 * a = e, where e is the identity.

For example, the set of integers forms a group under the operation of addition. It can be verified that for any three integers (such as 5, 7, and 3), their addition is associative, respectively, (5 + 7) + 3 = 12 + 3 = 15 = 5 + = 5 + (7 + 3). To show associativity, one must either assume or prove that every ordered selection of three integers satisfies the associativity property. Generally, it is assumed in most problems such as this. The identity element is 0, since summing zero to any number leaves it unchanged (for example, 5 + 0 = 5). The inverse of any number x is the number -x, so that x - x = -x + x = 0. Thus, the integers under addition form a group.

The set of integers under multiplication do not form a group. Although multiplication is associative and 1 is the multiplicative identity (that is, 1 * A = A for all integers A), the inverse of an integer is not always an integer (for example, the inverse of 5 is 1/5 = 0.2, which is not an integer).

The set of rational numbers under addition form a group. As another example, the rational numbers (excluding zero) under multiplication form a group since associativity holds, the identity is 1, and the inverse of any nonzero fraction is the reciprocal fraction. In a similar way, it can be shown that real numbers and complex numbers can form groups with either addition or multiplication.

Not all groups involve infinite sets. The set of integers mod n, denoted Z_n, forms a group under modular addition, with the following operation: a⊕b=(a+b)mod n. For example, in Z₃, one can compute 2⊕2=4mod 3=1. As a descriptive tool, finite groups are often characterized by an operation table for Z₃:

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Each entry in this table is computed by taking the number on the left and the number on top and applying the group operation (like an addition table). Note that from the table, the identity (zero) and the inverses can be found. So, 0 is the inverse for 0, 1 for 2, and 2 for 1, respectively. The study of finite groups is quite distinct from the theory of infinite groups, often using different tools and methodologies.

Groups can be constructed from objects other than simple numbers. The set of vectors under addition form a group. In two dimensions, two vectors <a,b> and <c,d> can be added by computing <a+b,c+d>. This is an operation commonly used in physics, namely vector addition by components. The identity is <0,0> and the inverse of <a,b> is <-a,-b>.

The sets of matrices under addition can also constitute groups. The identity in this case is a matrix with all zero entries, and the inverse would be the matrix with coefficients whose signs are reversed. More important, several classes of nonsingular matrices form groups under matrix multiplication. Associativity can be proved algebraically by expanding out terms in a matrix multiplication. The identity matrix consists of zero entries everywhere except the diagonal entries a_kk, which are all ones. Numerous algorithms exist for computing the inverse of a matrix. From linear algebra, it is known that every matrix describes a linear transformation. Since matrices define linear transformations, sets of linear transformations under composition (really multiplication) form several significant groups.

Familiar examples in geometry are rotation groups. Imagining an equilateral triangle on a plane, let 0, 120, and 240 represent rotations (in degrees). These three rotations form a group under "angle-addition." For example, 240 + 240 = 480 = 360 + 120 = 0 + 120 = 120. The identity is 0, while 240 and 120 are inverses. These rotations can map the triangle into itself, somewhat like spinning a triangle until its vertices overlap where the vertices were originally. An infinite rotation group could likewise be constructed, but by mapping a circle into itself by rotations. Rotating a circle any number of degrees leaves it appearing unmoved.

Other geometric groups can be defined on the plane using rotations, reflections, and translation, without necessarily mapping an object into itself. Translation moves an object from one point to another. Reflection takes mirror images. Groups can be constructed by combining all three of these motions.

In more general circumstances, sets of mappings can comprise groups. Groups can be constructed from permutations (that is, reorderings or shuffles) of numbers 1 to n. Each permutation reorders numbers 1 through n. The group operation would be a composition of two shuffles (that is, applying one shuffle after another in succession). The identity shuffle leaves 1 through n in their numerical order. The inverse of a shuffle is the permutation that fixes (sorts) 1 through n again after a permutation disorganized them. For example, the permutation that maps 1 to 2, 2 to 3, and 3 to 1 can be written as (Multiple line equation(s) cannot be represented in ASCII text; please see PDF of this article if available.) This is an example of the composition of two permutations:

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Based on these concepts, the set

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under composition is a group of permutations. In this set,

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the first permutation is the identity and the second permutation is the inverse of the third one.

Through groups of matrices, it has been shown that linear transformations can form groups; nonlinear transformations can also form groups. As one renowned example, consider the set of six transformations: {x’=x, x’=1/x, x'=1-x, x'=1/-x'), x'=(x-1)/x, x'=x/(x-1)}. For example, transformation 3 says take x as input and replace it with the value 1-x. Using the operation of composition (that is, plugging one transformation into another), any two transformations will give another one in this set. For example, composing transformation 3 into transformation 2 gives transformation 4. Try composing any other two transformations and note how the answer will turn out to be one of the six above. This is known as a binary operation. The identity is transformation 1, x = x. As for inverses, note that transformation 2 is its own inverse. The inverse of transformation 3 is 3. The inverse of 1 is 1. With a bit more manipulation, the others have inverses in the set, also. Therefore, this set forms a group of transformations. Groups of transformations have an important use in physics.

One important concept in group theory is that of a subgroup. If a subset of a group also satisfies the group properties, then that subset is also a group. For example, the set of even integers forms a group under addition and therefore is a subgroup of the integers. Subgroups act as a key to understanding the internal structure of a group.

Another important concept is that of isomorphic groups. Phrased intuitively, two groups are isomorphic if they "behave" the same or if their tables have the same structure. Note that the group Z₃ and the group of rotations {0, 124, 240} actually have the same structure. Underneath the different numbers being used, the two groups are behaving the same way. Delving a little further, the group of three permutations given earlier is also isomorphic to this group.

Another tool to be considered occurs in the area called combinatorial group theory. A group is often defined by a presentation, which is an ordered pair consisting of elements called generators and rewriting rules called relators. The set of all finite sequences of generators (and inverses) are created and then, by inserting and deleting relators, collapsed into a set of unique representatives. Those representatives form the group being studied. Using presentations, groups can be defined in a very compact, synoptic manner.

Generators and relators are used in definitions and constructions in topology. From a topological structure, generators and relators can be constructed that define a group for that structure (called the fundamental group). This fundamental group characterizes the topological object. In this manner, a topological problem can be reduced to a group theoretic one. Since several algorithms have been devised for presentations, problems in topology can be handled by computer. Since several problems in chemistry and physics are of a topological nature, this method is practical.

Applications

Groups of transformations have had a profound influence on physics, covertly operating during the period of Newtonian physics but visibly, formally, and more significantly under the relativistic model of Albert Einstein. As an example, an experiment can be conducted with pulleys and weights in a physics laboratory today. Repeat that experiment tomorrow. Repeat it on an adjacent table. Finally, repeat it after rotating the table. The behavior of the system and the results are the same. These tests have just shown that the physical laws are invariant under translation (moving to another table), rotation (turning tables), and translation in time. Rotations and translations form a group of transformations. Taking "snapshots" of each experiment, it is noted that these transformations define a symmetry between "snapshots." Under the Einsteinian model, the transformations are the famous Lorentz group. The Newtonian absolute time and absolute space was the basis for an alternate set of transformations, which appeared sufficient for the universe of Sir Isaac Newton's day. By accepting a different set of transformations, the geometric model of the cosmos has changed and, with it, twentieth century physics has also changed.

On the other extreme in scale, group theory has permeated much of quantum mechanics. The first comprehensive treatise on the manifold links between these disciplines is Hermann Weyl's THE THEORY OF GROUPS AND QUANTUM MECHANICS (1931). While much of it is beyond the scope of this presentation, Weyl masterfully applies group theory to the problems of spin, multiplets, the exclusion principle, and numerous other areas.

The power of groups of symmetries in particle physics can be noted in the application of the matrix group SU. Working with this group, physicists anticipated the omega minus baryon, which was later detected. Models of symmetries and their relations to particles occur in several texts, such as Murray Gell-Mann and Yuval Ne'eman's THE EIGHTFOLD WAY (1964). Yet, theories in this area are continually changing and are being constantly upgraded by observations from particle experiments.

Symmetries are also a required tool in modeling and investigating molecular systems in chemistry. A crystal is a recurrent structure of atoms. In studying crystals, three-dimensional rotation groups are required. As an example, imagine a spinning ball; it can be caused to spin in many possible ways. No matter how it is spun, it can be guaranteed that the center of the ball remains fixed (that is, the center never moves). This is called point symmetry. There are infinitely many possible rotations constituting the point symmetry group for a sphere because it can be rotated in any "direction" without constraint. Rotations with a crystal are more restrictive. As in the triangle example earlier, only rotations must be permitted which map the crystal's vertices to where the original state of the crystal had vertices. For example, if there were a prism with a triangle face up, then the prism could be rotated like the triangle, giving three rotations. If the prism is flipped upside down (so the lower triangle face is now the upper triangle face), and then rotated, then there are three more rotations (really three flip-and-rotations). After adding this up, it is clear that the prism has a point symmetry group of six elements.

By considering rotations of a crystal around a point, crystals can be classified according to their group. Chemists use other symmetries for classification as well, such as inversion, reflection, and combinations of all three. When taken all into account, these produce large numbers of classifications for molecular symmetry. While molecules in general can have any of these symmetries, the subset termed "crystals" can have only one of thirty-two groups (called crystal classes). Looking further, classification can occur with other symmetries that are not point symmetries (such as translation symmetries), which then constitute space groups, the very core of crystallography. These can be used in the analysis and interpretation of X-ray patterns.

Often in chemistry (and in physics), symmetries assist in the analysis of vibration. In studying vibration, the group of symmetries may help predict the modes of vibration. The Lagrange equations can then be formulated, solved, and the modes of vibration formally determined. These symmetries also make it possible to reduce the matrices and consequent problems in the process of computing the exact modes. In essence, symmetries give insight and simplify problems otherwise left to determinants of large matrices.

The approach of using transformation groups to simplify the treatment of differential equations resulted from Sophus Lie's work. The original intent was to achieve for differential equations the success that groups had with algebraic equations. The relationship was not the same, but it was still a triumph. Since chemistry, physics, mathematics, and most other sciences utilize differential equations in modeling many phenomena, this approach has reaped widespread benefits.

In computer science, group theory has always had an intimate affiliation. Several encryption schemes, error-correcting codes, modular random number generators, circuit designs, and theoretical computer models called automata have profited from several group theoretic principles.

George Polya formulated and applied group theoretic principles (the Polya-Burnside theorem) to the combinatorics of molecules. The theorem computes the number of unique compounds that can be created by adding or deleting radicals to molecules, one example of which was a benzene ring. The difficult issue was that some cases reduced to each other under symmetries. To avoid the redundant cases, those groups operating must be taken into account, as the formula does.

Context

The rudimentary concepts of sets date back to prehistory and are treated as a necessary semantic component in linguistics. As logic evolved, sets remained unnoticed, although they are directly related. Formal Boolean algebra originated with George Boole and his contemporaries in the early nineteenth century. Despite the increasing momentum of mathematical discoveries in the late 1800's, Georg Cantor's formulation of modern set theory was greeted with indifference and suspicion by some academic circles of that era. Opposing axiomatic systems and their corresponding schools developed. Nevertheless, set theory became firmly established in the twentieth century, with applications in every mathematical and scientific discipline. It is now a tenet of general education.

Symmetry has always fascinated geometricians, philosophers, and scientists. The formal groundwork for group theory was formulated by Joseph-Louis Lagrange. In his analysis of algebraic equations, Lagrange discovered the symmetric functions for expressing coefficients in terms of roots, concepts relating permutations of roots to solvability, and factorization by roots. Several elementary but essential theorems of group theory resulted from his efforts, most notably that the number of elements in a group is divisible by the number of elements in its subgroups.

The theory of numbers modulo n was formulated by Leonhard Euler and furthered by Carl Friedrich Gauss. In the nineteenth century, Evariste Galois' work on the algebraic-unsolvability of the general fifth and higher-degree algebraic equations spawned the concepts of normal subgroups and extensions. While Galois created the term "group" and Augustin-Louis Cauchy published the first definition of groups, Arthur Cayley was the first to publish the abstract definition. Cayley and his associates also developed matrix theory as an algebraic structure.

The intrinsically group theoretic nature of geometry was realized and expounded by Christian Felix Klein and Lie. Lie's work defined and established an extensive theory of transformation groups. It was the memorable work of Klein that interpreted group theory as an encompassing theme of geometry. Klein's goal (the Erlanger Programm) was a reassessment of geometries and geometric structures in terms of characteristics that are invariant under groups of transformations. For example, as one aspect of Euclidean geometry, areas are preserved under rotation.

Eventually, developments in transformation groups set the framework for the formulation of Einstein's theory of relativity, using the Lorentz group as the transformations permitted on the coordinates of the "real world."

In combinatorial group theory, topologists and group theorists from the mid-nineteenth century to the present developed related theories in a unified manner. The central problems of groups defined by presentations were developed by Max Dehn in the early 1900's. Dehn's problems would later be directly translated into algorithms for computers, and deep problems would be studied in a more systematic manner.

Designing new algorithms for solving group theoretic problems is an important direction of research. Computer packages for analyzing groups are one of the products of these efforts.

Finite groups are another area of great interest. Work in the classification of all finite groups is progressing. Simple groups are a special class of groups that serve as the building blocks to all finite groups. In the last quarter of the twentieth century, it is believed (with extensive evidence and proofs) that all finite simple groups have been constructively discovered and specified in well-defined categories.

Principal terms:

ELEMENT: an object that is a member of a set

INTEGER: any whole number that is either positive, negative, or zero; similarly, any number computed from zero by a finite number of additions of +1 or -1

INVERTIBLE MATRIX: a matrix whose inverse matrix can be computed; equivalently called nonsingular

MAPPING: an association (that is, function f) in which each element x of one set A has another corresponding unique element f(x) in set b; f is called a mapping and f maps A into b

MODULO: the operation (A mod b) is the remainder computed from the integer division of A by b, so: 17 mod 5 = 2; abbreviated mod

RATIONAL NUMBER: any number expressible as the quotient of two integers, a/b, where b is not zero

Z SUB N : the set {O, 1, 2, . . . , n - 1}, commonly called the integers modulo n

Bibliography

Aleksandrov, A. D., A. N. Kolmogorov, and M. A. Lavrent'ev. MATHEMATICS: ITS CONTENT, METHODS, AND MEANING. 3 vols. Cambridge, Mass.: MIT Press, 1963. The full breadth and wonder of mathematics is captured in this lucid exposition. No exercises are provided, but the focus is on comprehension rather than methods. Delightful coverage of classical mathematics and applications.

Buden, F. J. THE FASCINATION OF GROUPS. Cambridge, England: Cambridge University Press, 1972. This panorama of basic group theory is replete with thoroughly explained examples, completely worked out tables, and splendid illustrations. Very readable, clear, and intuitive, this book and Holland's (see below) coincide in applying group theory to bell ringing and spatial symmetries. Recommended.

Hargittai, Istvan, and Magdolna Hargittai. SYMMETRY THROUGH THE EYES OF A CHEMIST. New York: VCH, 1987. This text is particularly inspiring. Numerous photographs of symmetries in nature enhance the intuitive exposition. Exotic concepts such as galactic symmetry, snow crystals, elegant works of art, and a highly readable, refined style make this text irresistible as an introduction. The mathematical concepts are introduced in a nonoverwhelming manner. Its style is more like an architectural view of the universe than a textbook exercise drills. While chemistry is stressed, the goal is illumination, not methods and calculations alone. The authors thoroughly explain each point developed. The best introduction to applications of group theory to chemistry.

Hilbert, David. GEOMETRY AND THE IMAGINATION. New York: Chelsea, 1952. This book is a translation of a classic introduction to geometry as a subject to fire the imagination, written by one of the twentieth century's greatest mathematicians. In his brilliant yet readable style, Hilbert presents the basics of groups and their applications to crystallographic classes, lattices, and sphere packing in chapter 2. Fine illustrations.

Holland, Joan M. STUDIES IN STRUCTURE. London: Macmillan, 1972. This is among the easier introductions to groups applied to structures and symmetries, but devoid of advanced topics. Excellent illustrations and fascinating connections are developed with the Fibonacci series. Finite groups are characterized by an operation table for Z3

Group Theory and Elementary Particles