Free group
In mathematics, the free group FS over a given set S consists of all expressions that can be built from members of S, considering two expressions different unless their equality follows from the group axioms. The members of S are called generators of FS. An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in one and only one way as a product of finitely many elements of S and their inverses. A related but different notion is a free abelian group, both notions are particular instances of a free object from universal algebra. Free groups first arose as examples of Fuchsian groups. In an 1882 paper, Walther von Dyck pointed out that these groups have the simplest possible presentations; the algebraic study of free groups was initiated by Jakob Nielsen in 1924, who gave them their name and established many of their basic properties. Max Dehn realized the connection with topology, obtained the first proof of the full Nielsen–Schreier theorem.
Otto Schreier published an algebraic proof of this result in 1927, Kurt Reidemeister included a comprehensive treatment of free groups in his 1932 book on combinatorial topology. On in the 1930s, Wilhelm Magnus discovered the connection between the lower central series of free groups and free Lie algebras; the group of integers is free. A free group on a two-element set S occurs in the proof of the Banach–Tarski paradox and is described there. On the other hand, any nontrivial finite group cannot be free, since the elements of a free generating set of a free group have infinite order. In algebraic topology, the fundamental group of a bouquet of k circles is the free group on a set of k elements; the free group FS with free generating set. S is a set of symbols, we suppose for every s in S there is a corresponding "inverse" symbol, s−1, in a set S−1. Let T = S ∪ S−1, define a word in S to be any written product of elements of T; that is, a word in S is an element of the monoid generated by T. The empty word is the word with no symbols at all.
For example, if S = T =, a b 3 c − 1 c a − 1 c is a word in S. If an element of S lies next to its inverse, the word may be simplified by omitting the c, c−1 pair: a b 3 c − 1 c a − 1 c ⟶ a b 3 a − 1 c. A word that cannot be simplified further is called reduced; the free group FS is defined to be the group of all reduced words in S, with concatenation of words as group operation. The identity is the empty word. A word is called cyclically reduced, if last letter are not inverse to each other; every word is conjugate to a cyclically reduced word, a cyclically reduced conjugate of a cyclically reduced word is a cyclic permutation of the letters in the word. For instance b−1abcb is not cyclically reduced, but is conjugate to abc, cyclically reduced; the only cyclically reduced conjugates of abc are abc and cab. The free group FS is the universal group generated by the set S; this can be formalized by the following universal property: given any function ƒ from S to a group G, there exists a unique homomorphism φ: FS → G making the following diagram commute: That is, homomorphisms FS → G are in one-to-one correspondence with functions S → G.
For a non-free group, the presence of relations would restrict the possible images of the generators under a homomorphism. To see how this relates to the constructive definition, think of the mapping from S to FS as sending each symbol to a word consisting of that symbol. To construct φ for given ƒ, first note that φ sends the empty word to the identity of G and it has to agree with ƒ on the elements of S. For the remaining words φ can be uniquely extended since it is a homomorphism, i.e. φ = φ φ. The above property characterizes free groups up to isomorphism, is sometimes used as an alternative definition, it is known as the universal property of free groups, the generating set S is called a basis for FS. The basis for a free group is not uniquely determined. Being characterized by a universal property is the standard feature of free objects in universal algebra. In the language of category theory, the construction of the free group is a functor from the category of sets to the category of groups.
This functor is left adjoint to the forgetful functor from groups to sets. Some properties of free groups follow from the definition: Any group G is the homomorphic image of some free group F. Let S be a set of generators of G; the natural map f: F → G is an epimorphism, which proves the claim. Equivalently, G is isomorphic to a quotient group of some free group F; the kernel of φ is a set of relations in the presentation of G. If S can be chosen to be finite here G is called finitely generated. If S has more than one element F is not abelian, in fact the
Nilpotent group
A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with. In group theory, a nilpotent group is a group, "almost abelian"; this idea is motivated by the fact that nilpotent groups are solvable, for finite nilpotent groups, two elements having prime orders must commute. It is true that finite nilpotent groups are supersolvable; the concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov. Nilpotent groups arise in Galois theory, as well as in the classification of groups, they appear prominently in the classification of Lie groups. Analogous terms are used for Lie algebras including nilpotent, lower central series, upper central series; the definition uses the idea of a central series for a group. The following are equivalent definitions for a nilpotent group G: G has a central series of finite length; that is, a series of normal subgroups = G 0 ◃ G 1 ◃ ⋯ ◃ G n = G where G i + 1 / G i ≤ Z, or equivalently ≤ G i.
G has a lower central series terminating in the trivial subgroup after finitely many steps. That is, a series of normal subgroups G = G 0 ▹ G 1 ▹ ⋯ ▹ G n = where G i + 1 =. G has an upper central series terminating in the whole group after finitely many steps; that is, a series of normal subgroups = Z 0 ◃ Z 1 ◃ ⋯ ◃ Z n = G where Z 1 = Z and Z i + 1 is the subgroup such that Z i + 1 / Z i = Z. For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G. Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series. If a group has nilpotency class at most n it is sometimes called a nil-n group, it follows from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class 0, groups of nilpotency class 1 are the non-trivial abelian groups. As noted above, every abelian group is nilpotent. For a small non-abelian example, consider the quaternion group Q8, a smallest non-abelian p-group.
It has center of order 2, its upper central series is, Q8. The direct product of two nilpotent groups is nilpotent. All finite p-groups are in fact nilpotent; the maximal class of a group of order pn is n. The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, the semidihedral groups. Furthermore, every finite nilpotent group is the direct product of p-groups; the Heisenberg group H is an example of infinite nilpotent group. It has nilpotency class 2 with central series 1, Z, H; the multiplicative group of upper unitriangular n x n matrices over any field F is a nilpotent group of nilpotency class n - 1. The multiplicative group of invertible upper triangular n x n matrices over a field F is not in general nilpotent, but is solvable. Any nonabelian group G such that G/Z is abelian has nilpotency class 2, with central series, Z, G. Nilpotent groups are so called because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group G of nilpotence degree n and an element g, the function ad g: G → G defined by ad g := (where = g − 1 x − 1 g x {\
Cyclic group
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
Dihedral group
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 =
E8 (mathematics)
In mathematics, E8 is any of several related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, five exceptional cases labeled E6, E7, E8, F4, G2; the E8 algebra is the most complicated of these exceptional cases. The Lie group E8 has dimension 248, its rank, the dimension of its maximal torus, is eight. Therefore, the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly in this article; the Weyl group of E8, the group of symmetries of the maximal torus which are induced by conjugations in the whole group, has order 214 35 52 7 = 696729600. The compact group E8 is unique among simple compact Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation acting on the Lie algebra E8 itself. There is a Lie algebra Ek for every integer k ≥ 3, infinite dimensional if k is greater than 8.
There is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimension 248. The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496; this is connected, has maximal compact subgroup the compact form of E8, has an outer automorphism group of order 2 generated by complex conjugation. As well as the complex Lie group of type E8, there are three real forms of the Lie algebra, three real forms of the group with trivial center, all of real dimension 248, as follows: The compact form, connected and has trivial outer automorphism group; the split form, EVIII, which has maximal compact subgroup Spin/, fundamental group of order 2 and has trivial outer automorphism group. EIX, which has maximal compact subgroup E7×SU/, fundamental group of order 2 and has trivial outer automorphism group. For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups. By means of a Chevalley basis for the Lie algebra, one can define E8 as a linear algebraic group over the integers and over any commutative ring and in particular over any field: this defines the so-called split form of E8.
Over an algebraically closed field, this is the only form. Over R, the real connected component of the identity of these algebraically twisted forms of E8 coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all forms of E8 are connected in the sense of algebraic geometry, meaning that they admit no non-trivial algebraic coverings. Over finite fields, the Lang–Steinberg theorem implies that H1=0, meaning that E8 has no twisted forms: see below; the characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are: 1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250, 4825673125, 6899079264, 8634368000, 12692520960…The 248-dimensional representation is the adjoint representation.
There are two non-isomorphic irreducible representations of dimension 8634368000. The fundamental representations are those with dimensions 3875, 6696000, 6899079264, 146325270, 2450240, 30380, 248 and 147250; the coefficients of the character formulas for infinite dimensional irreducible representations of E8 depend on some large square matrices consisting of polynomials, the Lusztig–Vogan polynomials, an analogue of Kazhdan–Lusztig polynomials introduced for reductive groups in general by George Lusztig and David Kazhdan. The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations with the irreducible representations; these matrices were computed after four years of collaboration by a group of 18 mathematicians and computer scientists, led by Jeffrey Adams, with much of the programming done by Fokko du Cloux. The most difficult case is
Modular group
In mathematics, the modular group is the projective special linear group PSL of 2 x 2 matrices with integer coefficients and unit determinant. The matrices A and -A are identified; the modular group acts on the upper-half of the complex plane by fractional linear transformations, the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane which have the form z ↦ a z + b c z + d where a, b, c, d are integers, ad − bc = 1; the group operation is function composition. This group of transformations is isomorphic to the projective special linear group PSL, the quotient of the 2-dimensional special linear group SL over the integers by its center. In other words, PSL consists of all matrices where a, b, c, d are integers, ad − bc = 1, pairs of matrices A and −A are considered to be identical; the group operation is the usual multiplication of matrices. Some authors define the modular group to be PSL, still others define the modular group to be the larger group SL.
Some mathematical relations require the consideration of the group GL of matrices with determinant plus or minus one. PGL is the quotient group GL/. A 2 × 2 matrix with unit determinant is a symplectic matrix, thus SL = Sp, the symplectic group of 2x2 matrices; the unit determinant of implies that the fractions a/b, a/c, c/d and b/d are all irreducible, having no common factors. More if p/q is an irreducible fraction a p + b q c p + d q is irreducible. Any pair of irreducible fractions can be connected in this way, i.e.: for any pair p/q and r/s of irreducible fractions, there exist elements ∈ SL such that r = a p + b q and s = c p + d q. Elements of the modular group provide a symmetry on the two-dimensional lattice. Let ω 1 and ω 2 be two complex numbers whose ratio is not real; the set of points Λ = is a lattice of parallelograms on the plane. A different pair of vectors α 1 and α 2 will generate the same lattice if and only if = for some matrix in GL, it is for this reason that doubly periodic functions, such as elliptic functions, possess a modular group symmetry.
The action of the modular group on the rational numbers can most be understood by envisioning a square grid, with grid point corresponding to the fraction p/q. An irreducible fraction is one, visible from the origin. Note that any member of the modular group maps the projectively extended real line one-to-one to itself, furthermore bijectively maps the projectively extended rational line to itself, the irrationals to the irrationals, the transcendental numbers to the transcendental numbers, the non-real numbers to the non-real numbers, the upper half-plane to the upper half-plane, et cetera. If p n − 1 / q n − 1 and p n / q n are two successive convergents of a continued fraction the matrix (
Solenoid (mathematics)
This page discusses a class of topological groups. For the wrapped loop of wire, see Solenoid. In mathematics, a solenoid is a compact connected topological space that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms, fi: Si+1 → Si, i ≥ 0,where each Si is a circle and fi is the map that uniformly wraps the circle Si+1 ni times around the circle Si; this construction can be carried out geometrically in the three-dimensional Euclidean space R3. A solenoid is a one-dimensional homogeneous indecomposable continuum that has the structure of a compact topological group. In the special case where all ni have the same value n, so that the inverse system is determined by the multiplication by n self map of the circle, solenoids were first introduced by Vietoris for n = 2 and by van Dantzig for an arbitrary n; such a solenoid arises as a one-dimensional expanding attractor, or Smale–Williams attractor, forms an important example in the theory of hyperbolic dynamical systems.
Each solenoid may be constructed as the intersection of a nested system of embedded solid tori in R3. Fix a sequence of natural numbers, ni ≥ 2. Let T0 = S1 × D be a solid torus. For each i ≥ 0, choose a solid torus Ti+1, wrapped longitudinally ni times inside the solid torus Ti, their intersection Λ = ⋂ i ≥ 0 T i is homeomorphic to the solenoid constructed as the inverse limit of the system of circles with the maps determined by the sequence. Here is a variant of this construction isolated by Stephen Smale as an example of an expanding attractor in the theory of smooth dynamical systems. Denote the angular coordinate on the circle S1 by t and consider the complex coordinate z on the two-dimensional unit disk D. Let f be the map of the solid torus T = S1 × D into itself given by the explicit formula f =; this map is a smooth embedding of T into itself. If T is imagined as a rubber tube, the map f stretches it in the longitudinal direction, contracts each meridional disk, wraps the deformed tube twice inside T with twisting, but without self-intersections.
The hyperbolic set Λ of the discrete dynamical system is the intersection of the sequence of nested solid tori described above, where Ti is the image of T under the ith iteration of the map f. This set is a one-dimensional attractor, the dynamics of f on Λ has the following interesting properties: meridional disks are the stable manifolds, each of which intersects Λ over a Cantor set periodic points of f are dense in Λ the map f is topologically transitive on ΛGeneral theory of solenoids and expanding attractors, not one-dimensional, was developed by R. F. Williams and involves a projective system of infinitely many copies of a compact branched manifold in place of the circle, together with an expanding self-immersion. Solenoids are compact metrizable spaces that are connected, but not locally connected or path connected; this is reflected in their pathological behavior with respect to various homology theories, in contrast with the standard properties of homology for simplicial complexes.
In Čech homology, one can construct a non-exact long homology sequence using a solenoid. In Steenrod-style homology theories, the 0th homology group of a solenoid may have a complicated structure though a solenoid is a connected space. Protorus, a class of topological groups that includes the solenoids Pontryagin duality D. van Dantzig, Ueber topologisch homogene Kontinua, Fund. Math. 15, pp. 102–125 Hazewinkel, Michiel, ed. "Solenoid", Encyclopedia of Mathematics, Springer Science+Business Media B. V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Clark Robinson, Dynamical systems: Stability, Symbolic Dynamics and Chaos, 2nd edition, CRC Press, 1998 ISBN 978-0-8493-8495-0 S. Smale, Differentiable dynamical systems, Bull. of the AMS, 73, 747 – 817. L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97, pp. 454–472 Robert F. Williams, Expanding attractors, Publ. Math. IHES, t. 43, p. 169–203 Semmes, Some remarks about solenoids, arXiv:1201.2647, Bibcode:2012arXiv1201.2647S