In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, they play an important role in group theory and chemistry; the notation for the dihedral group differs in abstract algebra. In geometry, Dn or Dihn refers to the symmetries of a group of order 2n. In abstract algebra, D2n refers to this same dihedral group; the geometric convention is used in this article. A regular polygon with n sides has 2 n different symmetries: n rotational symmetries and n reflection symmetries. We take n ≥ 3 here; the associated rotations and reflections make up the dihedral group D n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.
The following picture shows the effect of the sixteen elements of D 8 on a stop sign: The first row shows the effect of the eight rotations, the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left. As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group; the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity. For example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°; the order of elements denoting the composition is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative. In general, the group Dn has elements r0, …, rn−1 and s0, …, sn−1, with composition given by the following formulae: r i r j = r i + j, r i s j = s i + j, s i r j = s i − j, s i s j = r i − j.
In all cases and subtraction of subscripts are to be performed using modular arithmetic with modulus n. If we center the regular polygon at the origin elements of the dihedral group act as linear transformations of the plane; this lets us represent elements of Dn with composition being matrix multiplication. This is an example of a group representation. For example, the elements of the group D4 can be represented by the following eight matrices: r 0 =, r 1 =, r 2 =, r 3 =, s 0 =, s 1 =, s 2 =
The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must be a finite group. Evgeny Golod and Igor Shafarevich provided a counter-example in 1964; the problem has many variants that differ in the additional conditions imposed on the orders of the group elements. Initial work pointed towards the affirmative answer. For example, if a group G is finitely generated and the order of each element of G is a divisor of 4 G is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists a largest one; this provides a solution for the restricted Burnside problem for the case of prime exponent. Issai Schur had showed in 1911 that any finitely generated periodic group, a subgroup of the group of invertible n × n complex matrices was finite; the general answer to Burnside problem turned out to be negative.
In 1964, Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order. In 1968, Pyotr Novikov and Sergei Adian's supplied a negative solution to the bounded exponent problem for all odd exponents larger than 4381. In 1982, A. Yu. Ol'shanskii found some striking counterexamples for sufficiently large odd exponents, supplied a simpler proof based on geometric ideas; the case of exponents turned out to be much harder to settle. In 1992, S. V. Ivanov announced the negative solution for sufficiently large exponents divisible by a large power of 2. Joint work of Ol'shanskii and Ivanov established a negative solution to an analogue of Burnside problem for hyperbolic groups, provided the exponent is sufficiently large. By contrast, when the exponent is small and different from 2,3,4 and 6 little is known. A group G is called periodic; every finite group is periodic. There exist defined groups such as the p∞-group which are infinite periodic groups.
General Burnside problem. If G is a finitely generated, periodic group is G finite? This question was answered in the negative in 1964 by Evgeny Golod and Igor Shafarevich, who gave an example of an infinite p-group, finitely generated. However, the orders of the elements of this group are not a priori bounded by a single constant. Part of the difficulty with the general Burnside problem is that the requirements of being finitely generated and periodic give little information about the possible structure of a group. Therefore, we pose more requirements on G. Consider a periodic group G with the additional property that there exists a least integer n such that for all g in G, gn = 1. A group with this property is said to be periodic with bounded exponent n, or just a group with exponent n. Burnside problem for groups with bounded exponent asks: Burnside problem. If G is a finitely generated group with exponent n, is G finite? It turns out that this problem can be restated as a question about the finiteness of groups in a particular family.
The free Burnside group of rank m and exponent n, denoted B, is a group with m distinguished generators x1... xm in which the identity xn = 1 holds for all elements x, and, the "largest" group satisfying these requirements. More the characteristic property of B is that, given any group G with m generators g1... gm and of exponent n, there is a unique homomorphism from B to G that maps the ith generator xi of B into the ith generator gi of G. In the language of group presentations, free Burnside group B has m generators x1... xm and the relations xn = 1 for each word x in x1... xm, any group G with m generators of exponent n is obtained from it by imposing additional relations. The existence of the free Burnside group and its uniqueness up to an isomorphism are established by standard techniques of group theory, thus if G is any finitely generated group of exponent n G is a homomorphic image of B, where m is the number of generators of G. Burnside problem can now be restated as follows: Burnside problem II.
For which positive integers m, n is the free Burnside group B finite? The full solution to Burnside problem in this form is not known. Burnside considered some easy cases in his original paper: B is the cyclic group of order n. B is the direct product of m copies of the cyclic group of order 2 and hence finite; the following additional results are known: B, B, B are finite for all m. The particular case of B remains open: as of 2005 it was not known whether this group is finite; the breakthrough in Burnside problem was achieved by Pyotr Novikov and Sergei Adian in 1968. Using a complicated combinatorial argument, they demonstrated that for every odd number n with n > 4381, there exist infinite, finitely generated groups of exponent n. Adian improved the bound on the odd exponent to 665; the case of exponent turned out to be more difficult. It was only in 1994 that Sergei Vasilievich Ivanov was able to prove an analogue of Novikov–Adian theorem: for any m > 1 and an n ≥ 248, n divisible by 29, the group B i
In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group, generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, it contains an element g such that every other element of the group may be obtained by applying the group operation to g or its inverse; each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group; every infinite cyclic group is isomorphic to the additive group of the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n; every cyclic group is an abelian group, every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order; the cyclic groups of prime order are thus among the building blocks from which all groups can be built.
For any element g in any group G, one can form the subgroup of all integer powers ⟨g⟩ =, called the cyclic subgroup of g. The order of g is the number of elements in ⟨g⟩. A cyclic group is a group, equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator. For a finite cyclic group with order |G| = n, this means G =, where e is the identity element and gj = gk whenever j ≡ k modulo n. An abstract group defined by this multiplication is denoted Cn, we say that G is isomorphic to the standard cyclic group Cn; such a group is isomorphic to Z/nZ, the group of integers modulo n with the addition operation, the standard cyclic group in additive notation. Under the isomorphism χ defined by χ = i the identity element e corresponds to 0, products correspond to sums, powers correspond to multiples. For example, the set of complex 6th roots of unity G = forms a group under multiplication, it is cyclic, since it is generated by the primitive root z = 1 2 + 3 2 i = e 2 π i / 6: that is, G = ⟨z⟩ = with z6 = 1.
Under a change of letters, this is isomorphic to the standard cyclic group of order 6, defined as C6 = ⟨g⟩ = with multiplication gj · gk = gj+k, so that g6 = g0 = e. These groups are isomorphic to Z/6Z = with the operation of addition modulo 6, with zk and gk corresponding to k. For example, 1 + 2 ≡ 3 corresponds to z1 · z2 = z3, 2 + 5 ≡ 1 corresponds to z2 · z5 = z7 = z1, so on. Any element generates its own cyclic subgroup, such as ⟨z2⟩ = of order 3, isomorphic to C3 and Z/3Z. Instead of the quotient notations Z/nZ, Z/, or Z/n, some authors denote a finite cyclic group as Zn, but this conflicts with the notation of number theory, where Zp denotes a p-adic number ring, or localization at a prime ideal. On the other hand, in an infinite cyclic group G = ⟨g⟩, the powers gk give distinct elements for all integers k, so that G =, G is isomorphic to the standard group C = C∞ and to Z, the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, the name "cyclic" may be misleading.
To avoid this confusion, Bourbaki introduced the term monogenous group for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group". The set of integers Z,with the operation of addition, forms a group, it is an infinite cyclic group, because all integers can be written by adding or subtracting the single number 1. In this group, 1 and −1 are the only generators; every infinite cyclic group is isomorphic to Z. For every positive integer n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group, denoted Z/nZ. A modular integer i is a generator of this group if i is prime to n, because these elements can generate all other elements of the group through integer addition; every finite cyclic group G is isomorphic to Z/nZ. The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of commutative rings denoted Z and Z/nZ or Z/.
If p is a prime Z/pZ is a finite field, is denoted Fp or GF. For every positive integer n, the set of the integers modulo n that are prime to n is written as ×; this group is not always cyclic, bu
Least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b denoted by LCM, is the smallest positive integer, divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define LCM as 0 for all a, the result of taking the LCM to be the least upper bound in the lattice of divisibility; the LCM is the "lowest common denominator" that can be used before fractions can be added, subtracted or compared. The LCM of more than two integers is well-defined: it is the smallest positive integer, divisible by each of them. A multiple of a number is the product of an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2; because 10 is the smallest positive integer, divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of − 2 as well.
In this article we will denote the least common multiple of two integers b as lcm. Some older textbooks use; the programming language J uses a*.b What is the LCM of 4 and 6? Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76...and the multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72... Common multiples of 4 and 6 are the numbers that are in both lists: 12, 24, 36, 48, 60, 72.... So, from this list of the first few common multiples of the numbers 4 and 6, their least common multiple is 12; when adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. For instance, 2 21 + 1 6 = 4 42 + 7 42 = 11 42 where the denominator 42 was used because it is the least common multiple of 21 and 6. Suppose there are two meshing gears in a machine, having m and n teeth and the gears are marked by a line segment drawn from the center of the first gear to the center of the second gear.
When the gears begin rotating, we can determine how many rotations the first gear must complete to realign the line segment by making use of LCM. The first gear must complete LCM/m rotations for the realignment. By that time, the second gear will have made LCM/n rotations. Suppose there are three planets revolving around a star which take l, m and n units of time to complete their orbits. Assume that l, m and n are integers. Assuming the planets started moving around the star after an initial linear alignment, all the planets attain a linear alignment again after LCM units of time. At this time, the first and third planet will have completed LCM/l, LCM/m and LCM/n orbits around the star; the following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor known as the greatest common factor: lcm = | a ⋅ b | gcd . This formula is valid when one of a and b is 0, since gcd = |a|. However, if both a and b are 0, this formula would cause division by zero.
There are fast algorithms for computing the GCD that do not require the numbers to be factored, such as the Euclidean algorithm. To return to the example above, lcm = 21 ⋅ 6 gcd = 21 ⋅ 6 gcd = 21 ⋅ 6 3 = 126 3 = 42; because gcd is a divisor of both a and b, it is more efficient to compute the LCM by dividing before multiplying: lcm = ⋅ | b | = ⋅ | a |. This reduces the size of one input for both the division and the multiplication, reduces the required storage needed for inte
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
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set. Writing A = means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example, are subsets of A. Sets can themselves be elements. For example, consider the set B =; the elements of B are not 1, 2, 3, 4. Rather, there are only three elements of B, namely the numbers 1 and 2, the set; the elements of a set can be anything. For example, C =, is the set whose elements are the colors red and blue; the relation "is an element of" called set membership, is denoted by the symbol " ∈ ". Writing x ∈ A means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A"; the expressions "A includes x" and "A contains x" are used to mean set membership, however some authors use them to mean instead "x is a subset of A". Logician George Boolos urged that "contains" be used for membership only and "includes" for the subset relation only.
For the relation ∈, the converse relation ∈T may be written A ∋ x, meaning "A contains x". The negation of set membership is denoted by the symbol "∉". Writing x ∉ A means that "x is not an element of A"; the symbol ∈ was first used by Giuseppe Peano 1889 in his work Arithmetices principia, nova methodo exposita. Here he wrote on page X: Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b. So a ∈ b is read; every relation R: U → V is subject to two involutions: complementation R → R ¯ and conversion RT: V → U. The relation ∈ has for its domain a universal set U, has the power set P for its codomain or range; the complementary relation ∈ ¯ = ∉ expresses the opposite of ∈. An element x ∈ U may have x ∉ A, in which case x ∈ U \ A, the complement of A in U; the converse relation ∈ T = ∋ swaps the domain and range with ∈. For any A in P, A ∋ x is true when x ∈ A; the number of elements in a particular set is a property known as cardinality. In the above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3.
An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers. Using the sets defined above, namely A =, B = and C =: 2 ∈ A ∈ B 3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite and equal to 5; the cardinality of P = is infinite. Halmos, Paul R. Naive Set Theory, Undergraduate Texts in Mathematics, NY: Springer-Verlag, ISBN 0-387-90092-6 - "Naive" means that it is not axiomatized, not that it is silly or easy. Jech, Thomas, "Set Theory", Stanford Encyclopedia of Philosophy Suppes, Axiomatic Set Theory, NY: Dover Publications, Inc. ISBN 0-486-61630-4 - Both the notion of set, membership or element-hood, the axiom of extension, the axiom of separation, the union axiom are needed for a more thorough understanding of "set element". Weisstein, Eric W. "Element". MathWorld