- The operation is associative.
- The operation has an identity element.
- Every element has an inverse element.
(Read on for more precise definitions.)
Groups are building blocks of more elaborate algebraic structures such as rings, fields, and vector spaces, and recur throughout mathematics. Group theory has many applications in physics and chemistry, and is potentially applicable in any situation characterized by symmetry.
The order of a group is the cardinality of G; groups can be of finite or infinite order. The classification of finite simple groups is a major mathematical achievement of the 20th century.
Group theory concepts
A group consists of a collection of abstract objects or symbols, and a rule for combining them. The combination rule indicates how these objects are to be manipulated. Hence groups are a way of doing mathematics with symbols instead of concrete numbers.
More precisely, one may speak of a group whenever a set, together with an operation that always combines two elements of this set, for example, a x b, always fulfills the following requirements:
- The combination of two elements of the set yields an element of the same set ( closure);
- The bracketing is unimportant (associativity): a × (b × c) = (a × b) × c;
- There is an element that does not cause anything to happen ( identity element): a × 1 = 1 × a = a;
- Each element a has a "mirror image" ( inverse element) 1/a that has the property to yield the identity element when combined with a: a × 1/a = 1/a × a = 1
Special case: If the order of the operands does not affect the result, that is if a × b = b × a holds (commutativity), then we speak of an abelian group.
Some simple numeric examples of abelian groups are:
- Integers with the addition operation "+" as binary operation and zero as identity element
- Rational numbers without zero with multiplication "x" as binary operation and the number one as identity element. Zero has to be excluded because it does not have an inverse element. ("1/0" is undefined.)
This definition of groups is deliberately very general. It allows one to treat as groups not only sets of numbers with corresponding operations, but also other abstract objects and symbols that fulfill the required properties, such as polygons with their rotations and reflections in dihedral groups.
James Newman summarized group theory as follows:
|“||The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing.||”|
Definition of a group
A group (G, *) is a set G closed under a binary operation * satisfying the following 3 axioms:
- Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
- Identity element: There exists an e∈G such that for all a in G, e * a = a * e = a.
- Inverse element: For each a in G, there is an element b in G such that a * b = b * a = e, where e is the identity element.
In the terminology of universal algebra, a group is a variety, and a algebra of type .
A set H is a subgroup of a group G if it is a subset of G and is a group using the operation defined on G. In other words, H is a subgroup of (G, *) if the restriction of * to H is a group operation on H.
A subgroup H is a normal subgroup of G if for all h in H and g in G, ghg-1 is also in H. An alternative (but equivalent) definition is that a subgroup is normal if its left and right cosets coincide. Normal subgroups play a distinguished role by virtue of the fact that the collection of cosets of a normal subgroup N in a group G naturally inherits a group structure, enabling the formation of the quotient group, usually denoted G/N (also sometimes called a factor group).
Operations involving groups
A homomorphism is a map between two groups that preserves the structure imposed by the operator. If the map is bijective, then it is an isomorphism. An isomorphism from a group to itself is an automorphism. The set of all automorphisms of a group is a group called the automorphism group. The kernel of a homomorphism is a normal subgroup of the group.
A group action is a map involving a group and a set, where each element in the group defines a bijective map on a set. Group actions are used to prove the Sylow theorems and to prove that the centre of a p-group is nontrivial.
Special types of groups
A group is:
- Abelian (or commutative) if its product commutes (that is, for all a, b in G, a * b = b * a). A non-abelian group is a group that is not abelian. The term "abelian" honours the mathematician Niels Abel.
- Cyclic if it is generated by a single element.
- Simple if it has no nontrivial normal subgroups.
- Solvable (or soluble) if it has a normal series whose quotient groups are all abelian. The fact that S5, the symmetric group in 5 elements, is not solvable is used to prove that some quintic polynomials cannot be solved by radicals.
- Free if there exists a subset of G, H, such that all elements of G can be written uniquely as products (or strings) of elements of H. Every group is the homomorphic image of some free group.
Some useful theorems
Some basic results in elementary group theory:
- Lagrange's theorem: if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G.
- Cayley's Theorem: every group G is isomorphic to a subgroup of the symmetric group on G.
- Sylow theorems: if pn (and p prime) is the greatest power of p dividing the order of a finite group G, then there exists a subgroup of order pn. This is perhaps the most useful basic result on finite groups.
- The Butterfly lemma is a technical result on the lattice of subgroups of a group.
- The Fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
- Jordan-Hölder theorem: any two composition series of a given group are equivalent.
- Krull-Schmidt theorem: a group G satisfying certain finiteness conditions for chains of its subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
- Burnside's lemma: the number of orbits of a group action on a set equals the average number of points fixed by each element of the group.
Connection of groups and symmetry
Given a structured object of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. For example rotations of a sphere are symmetries of the sphere. If the object is a set with no additional structure, a symmetry is a bijective map from the set to itself. If the object is a set of points in the plane with its metric structure, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry).
The axioms of a group formalize the essential aspects of symmetry.
- Closure of the group law - This says if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry.
- The existence of an identity - This says that keeping the object fixed is always a symmetry of an object.
- The existence of inverses - This says every symmetry can be undone.
- Associativity - Since symmetries are functions on a space, and composition of functions are associative, this axiom is needed to make a formal group behave like functions.
Frucht's theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object.
Applications of group theory
Some important applications of group theory include:
- Groups are often used to capture the internal symmetry of other structures. An internal symmetry of a structure is usually associated with an invariant property; the set of transformations that preserve this invariant property, together with the operation of composition of transformations, form a group called a symmetry group. Also see automorphism group.
- Galois theory, which is the historical origin of the group concept, uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The solvable groups are so-named because of their prominent role in this theory. Galois theory was originally used to prove that polynomials of the fifth degree and higher cannot, in general, be solved in closed form by radicals, the way polynomials of lower degree can.
- Abelian groups, which add the commutative property a * b = b * a, underlie several other structures in abstract algebra, such as rings, fields, and modules.
- In algebraic topology, groups are used to describe invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. Examples include the fundamental group, homology groups and cohomology groups. The name of the torsion subgroup of an infinite group shows the legacy of topology in group theory.
- The concept of the Lie group (named after mathematician Sophus Lie) is important in the study of differential equations and manifolds; they describe the symmetries of continuous geometric and analytical structures. Analysis on these and other groups is called harmonic analysis.
- In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma.
- An understanding of group theory is also important in physics and chemistry and material science. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include: Standard Model, Gauge theory, Lorentz group, Poincaré group
- In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and Infrared spectroscopy), and to construct molecular orbitals.
- Group theory is used extensively in public-key cryptography. In Elliptic-Curve Cryptography, very large groups of prime order are constructed by defining elliptic curves over finite fields.
There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Euler, Gauss, Lagrange, Abel and French mathematician Galois were early researchers in the field of group theory. Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory.
An early source occurs in the problem of forming an th-degree equation having as its roots m of the roots of a given th-degree equation (). For simple cases the problem goes back to Hudde (1659). Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Le Sœur (1748) and Waring (1762 to 1782) still further elaborated the idea.
A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. He discovered that the roots of all resolvents (résolvantes, réduites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a Calcul des Combinaisons. The contemporary work of Vandermonde (1770) also foreshadowed the coming theory.
Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni. He also published a letter from Abbati to himself, in which the group idea is prominent.
Galois found that if are the roots of an equation, there is always a group of permutations of the 's such that (1) every function of the roots invariable by the substitutions of the group is rationally known, and (2), conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI).
Arthur Cayley and Augustin Louis Cauchy were among the first to appreciate the importance of the theory, and to the latter especially are due a number of important theorems. The subject was popularised by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des Substitutions is a classic; and to Eugen Netto (1882), whose Theory of Substitutions and its Applications to Algebra was translated into English by Cole (1892). Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Emile Mathieu.
Walther von Dyck was the first (in 1882) to define a group in the full abstract sense of this entry.
The study of what are now called Lie groups, and their discrete subgroups, as transformation groups, started systematically in 1884 with Sophus Lie; followed by work of Killing, Study, Schur, Maurer, and Cartan. The discontinuous ( discrete group) theory was built up by Felix Klein, Lie, Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy.
The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.
Other important contributors to group theory include Emil Artin, Emmy Noether, Sylow, and many others.
Alfred Tarski proved elementary group theory undecidable.
An application of group theory is musical set theory.
In philosophy, Ernst Cassirer related group theory to the theory of perception of Gestalt Psychology. He took the Perceptual Constancy of that psychology as analogous to the invariants of group theory.