Group (mathematics)

In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.

After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.

The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.

Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.

Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.

For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words

Fundamental theorem of arithmetic

In number theory, the fundamental theorem of arithmetic called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, this representation is unique, up to the order of the factors. For example, 1200 = 24 × 31 × 52 = 2 × 2 × 2 × 2 × 3 × 5 × 5 = 5 × 2 × 5 × 2 × 3 × 2 × 2 =... The theorem says two things for this example: first, that 1200 can be represented as a product of primes, second, that no matter how this is done, there will always be four 2s, one 3, two 5s, no other primes in the product; the requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique. This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime factorization into primes would not be unique. Book VII, propositions 30, 31 and 32, Book IX, proposition 14 of Euclid's Elements are the statement and proof of the fundamental theorem.

If two numbers by multiplying one another make some number, any prime number measure the product, it will measure one of the original numbers. Proposition 30 is referred to as Euclid's lemma, it is the key in the proof of the fundamental theorem of arithmetic. Any composite number is measured by some prime number. Proposition 31 is proved directly by infinite descent. Any number either is measured by some prime number. Proposition 32 is derived from proposition 31, proves that the decomposition is possible. If a number be the least, measured by prime numbers, it will not be measured by any other prime number except those measuring it. Book IX, proposition 14 is derived from Book VII, proposition 30, proves that the decomposition is unique – a point critically noted by André Weil. Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.

Every positive integer n > 1 can be represented in one way as a product of prime powers: n = p 1 n 1 p 2 n 2 ⋯ p k n k = ∏ i = 1 k p i n i where p1 < p2 <... < pk are primes and the ni are positive integers. This representation is extended to all positive integers, including 1, by the convention that the empty product is equal to 1; this representation is called the canonical representation of n, or the standard form of n. For example, 999 = 33×37, 1000 = 23×53, 1001 = 7×11×13. Note that factors p0 = 1 may be inserted without changing the value of n. In fact, any positive integer can be uniquely represented as an infinite product taken over all the positive prime numbers: n = 2 n 1 3 n 2 5 n 3 7 n 4 ⋯ = ∏ i = 1 ∞ p i n i, where a finite number of the ni are positive integers, the rest are zero. Allowing negative exponents provides a canonical form for positive rational numbers; the canonical representations of the product, greatest common divisor, least common multiple of two numbers a and b can be expressed in terms of the canonical representations of a and b themselves: a ⋅ b = 2 a 1 + b 1 3 a 2 + b 2 5 a 3 + b 3 7 a 4 + b 4 ⋯ = ∏ p i a i + b i, gcd = 2 min 3 min ( a 2, b 2

Isomorphism theorems

In mathematics abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients and subobjects. Versions of the theorems exist for groups, vector spaces, Lie algebras, various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences; the isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether. Three years B. L. van der Waerden published his influential Algebra, the first abstract algebra textbook that took the groups-rings-fields approach to the subject. Van der Waerden credited lectures by Noether on group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, Otto Schreier, van der Waerden himself on ideals as the main references.

The three isomorphism theorems, called homomorphism theorem, two laws of isomorphism when applied to groups, appear explicitly. We first state the three isomorphism theorems in the context of groups. Note that some sources switch the numbering of the second and third theorems. Another variation encountered in the literature in Van der Waerden's Algebra, is to call first isomorphism theorem the Fundamental Homomorphism Theorem and to decrement the numbering of the remaining isomorphism theorems by one. In the most extensive numbering scheme, the lattice theorem is sometimes referred to as the fourth isomorphism theorem. Let G and H be groups, let φ: G → H be a homomorphism. Then: The kernel of φ is a normal subgroup of G, The image of φ is a subgroup of H, The image of φ is isomorphic to the quotient group G / ker. In particular, if φ is surjective H is isomorphic to G / ker. Let G be a group. Let S be a subgroup of G, let N be a normal subgroup of G; the following hold: The product S N is a subgroup of G, The intersection S ∩ N is a normal subgroup of S, The quotient groups / N and S / are isomorphic.

Technically, it is not necessary for N to be a normal subgroup, as long as S is a subgroup of the normalizer of N in G. In this case, the intersection S ∩ N is not a normal subgroup of G, but it is still a normal subgroup of S; this isomorphism theorem has been called the "diamond theorem" due to the shape of the resulting subgroup lattice with S N at the top, S ∩ N at the bottom and with N and S to the sides. It has been called the "parallelogram theorem" because in the resulting subgroup lattice the two sides assumed to represent the quotient groups / N and S / are "equal" in the sense of isomorphism. An example of the second isomorphism theorem gives an identity of projective linear groups. Setting G = G L 2, the group of invertible 2x2 complex matrices, S = S L 2, the subgroup of determinant 1 matrices, N the normal subgroup of scalar matrices C × I =, we have S ∩ N =, where I is the identity matrix, S N = G L 2; the second isomorphism theorem states that: P G L 2:= G L 2 / ≅ S L 2 / =: P S L

Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series and functions. A ring is an abelian group with a second binary operation, associative, is distributive over the abelian group operation, has an identity element. By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication. Whether a ring is commutative or not has profound implications on its behavior as an abstract object; as a result, commutative ring theory known as commutative algebra, is a key topic in ring theory. Its development has been influenced by problems and ideas occurring in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, the ring of integers of a number field.

Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, the cohomology ring of a topological space in topology. The conceptualization of rings was completed in the 1920s. Key contributors include Dedekind, Hilbert and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry and mathematical analysis; the most familiar example of a ring is the set of all integers, Z, consisting of the numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms R is an abelian group under addition, meaning that: + c = a + for all a, b, c in R. a + b = b + a for all a, b in R.

There is an element 0 in R such that a + 0 = a for all a in R. For each a in R there exists −a in R such that a + = 0. R is a monoid under multiplication, meaning that: · c = a · for all a, b, c in R. There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R. Multiplication is distributive with respect to addition, meaning that: a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many authors follow an alternative convention in which a ring is not defined to have a multiplicative identity; this article adopts the convention that, unless otherwise stated, a ring is assumed to have such an identity. A structure satisfying all the axioms except the requirement that there exists a multiplicative identity element is called a rng. For example, the set of integers with the usual + and ⋅ is a rng, but not a ring; the operations + and ⋅ are called multiplication, respectively. The multiplication symbol ⋅ is omitted, so the juxtaposition of ring elements is interpreted as multiplication.

For example, xy means x ⋅ y. Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not equal ba. Rings that satisfy commutativity for multiplication are called commutative rings. Books on commutative algebra or algebraic geometry adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplication does not have to have an inverse. A commutative ring such; the additive group of a ring is the ring equipped just with the structure of addition. Although the definition assumes that the additive group is abelian, this can be inferred from the other ring axioms; some basic properties of a ring follow from the axioms: The additive identity, the additive inverse of each element, the multiplicative identity are unique. For any element x in a ring R, one has x0 = 0 = 0x and x = –x. If 0 = 1 in a ring R R has only one element, is called the zero ring; the binomial formula holds for any commuting pair of elements. Equip the set Z 4 = with the following operat