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Topic: Congruence subgroup

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	Congruence subgroup - Biocrawler (Site not responding. Last check: 2007-10-26)
		In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries.
		More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' respected by the subgroup, and so some general idea of what 'congruence' means.
		That is, it is the congruence subgroup that is the kernel of reduction modulo 2, otherwise known as Γ(2).
	www.biocrawler.com /encyclopedia/Congruence_subgroup (557 words)

	PlanetMath: quotient group
		Before defining quotient groups, some preliminary definitions must be introduced and a few propositions established.
		proof that a subgroup of a group defines an equivalence relation on the group
		Cross-references: dihedral group, center, cyclic, isomorphic, order, generated by, cyclic subgroup, integers, multiplication, disjoint, implies, hypothesis, normal subgroup, well-defined, bijection, cardinality, right coset, left coset, equivalence classes, mutually disjoint, binary operation, closure, symmetric, inverses, identity, contains, equivalence relation, right, abelian, relation, congruence, subgroup, group, definitions
	planetmath.org /encyclopedia/FactorGroup.html (461 words)

	Discrete group - Wikipedia, the free encyclopedia
		A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one.
		Every triangle group T is a discrete subgroup of the isometry group of the sphere (when T is finite), the Euclidean plane (when T has a Z + Z subgroup of finite index), or the hyperbolic plane.
		A lattice in a Lie group is a discrete subgroup such that the Haar measure of the quotient space is finite.
	www.wikipedia.org /wiki/Discrete_group (640 words)

	Congruence subgroup (Site not responding. Last check: 2007-10-26)
		In mathematics a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the A very simple example would be invertible 2x2 integer matrices of determinant 1 such that the off-diagonal entries even.
		The kernel of this reduction is an example of a congruence subgroup; case n=2 we are talking then about subgroup of the modular group (up to the quotient by {I taking us to the corresponding projective group): kernel of reduction is called then Γ(N) plays a big role in the theory modular forms.
		That is it is the congruence that is the kernel of reduction modulo otherwise known as Γ(2).
	www.freeglossary.com /Congruence_subgroup (595 words)

	Modular group - Wikipedia, the free encyclopedia
		The modular group is important because it forms a subgroup of the group of isometries of the hyperbolic plane.
		Important subgroups of the modular group Γ, called congruence subgroups, are given by imposing congruence relations on the associated matrices.
		The modular group and its subgroups were first studied in detail by Dedekind and by Felix Klein as part of his Erlangen programme in the 1870s.
	en.wikipedia.org /wiki/Modular_group (1319 words)

	Congruence subgroup - Wikipedia, the free encyclopedia
		It can be posed in topological terms: if Γ is some arithmetic group, there is a topology on Γ for which a base of neighbourhoods of {e} is the set of subgroups of finite index; and there is another topology defined in the same way using only congruence subgroups.
		The subgroups of finite index give rise to the completion of Γ as a pro-finite group.
		If there are essentially fewer congruence subgroups, the corresponding completion of Γ can be bigger (intuitively, there are fewer conditions for a Cauchy sequence to comply with).
	en.wikipedia.org /wiki/Congruence_subgroup (565 words)

	Congruence subgroup (Site not responding. Last check: 2007-10-26)
		More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraicgroups ; that is, those for which we have a notion of 'integral structure' respected by the subgroup, and so some general ideaof what 'congruence' means.
		The use of adele methods for automorphic representations (for example in the Langlands program) implicitly uses that kind of completion with respect to a congruence subgroup topology- for the reason that then all congruence subgroups can then be treated within a single group representation.
		That is, it is the congruence subgroup that is the kernel of reductionmodulo 2, otherwise known as Γ(2).
	www.therfcc.org /congruence-subgroup-218252.html (542 words)

	Congruence subgroup -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-26)
		It can be posed in topological terms: if Γ is some arithmetic group, there is a topology on Γ for which a base of neighbourhoods of is the set of subgroups of finite index; and there is another topology defined in the same way using only congruence subgroups.
		If there are essentially fewer congruence subgroups, the corresponding completion of Γ can be bigger (intuitively, there are fewer conditions for a (Click link for more info and facts about Cauchy sequence) Cauchy sequence to comply with).
		The modular group Λ (also called the theta subgroup) is another ((mathematics) a subset (that is not empty) of a mathematical group) subgroup of the modular group Γ.
	www.absoluteastronomy.com /encyclopedia/c/co/congruence_subgroup.htm (586 words)

	Congruence relation (Site not responding. Last check: 2007-10-26)
		In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraicoperation(s).
		Congruences typically arise as kernels of homomorphisms, and in fact every congruence is the kernel ofsome homomorphism: For a given congruence ~ on A, the set A/~ of equivalence classes can be given the structure of an algebra in a natural fashion, the quotient algebra.
		Notice that such a congruence ~ is determined entirely by the set {a ∈ G : a ~ e} ofthose elements of G that are congruent to the identity element, and this set is a normal subgroup.
	www.therfcc.org /congruence-relation-34856.html (460 words)

	Congruence Subgroups
		The genus of a subgroup U of Γ is the genus of the corresponding surface H/U. The principal congruence subgroup of level N, Γ(N), is the image in PSL(2,Z) of the group {[a,b,c,d] in SL(2,Z) with [a,b,c,d] = [1,0,0,1] mod N}.
		A subgroup of Γ which contains some principal congruence subgroup is called a congruence subgroup.
		We present complete tables of all congruence subgroups of PSL(2,Z) of of genus 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24.
	www.mathstat.concordia.ca /faculty/cummins/congruence (622 words)

	Math (Site not responding. Last check: 2007-10-26)
		Let A be the automorphism group of a regular tree and let G be the subgroup generated by k random elements in A. We show that almost surely G is free of rank k and every nonidentity element of G has 0 or 2 fixpoints on the boundary of the tree.
		We show that subgroups of Γ(p) generated by three random elements are full-dimensional and that there exist finitely generated subgroups of arbitrary dimension.
		This is applied to the determination of congruence subgroup growth of arithmetic groups over global fields of positive characteristic.
	www.math.uchicago.edu /~abert/research.html (753 words)

	[No title]
		Note that the computation of the metaplectic kernel is required for a solution of the congruence subgroup problem and also for the theory of automorphic forms of fractional weights.
		Whenever the congruence subgroup kernel is central, it is isomorphic to the dual of the metaplectic kernel.
		This formula was used in a subsequent paper [13], written jointly with Armand Borel, to prove the finiteness of the number of S-arithmetic subgroups with covolume bounded by a given number and also the finiteness of number of groups of compact type with a given class number.
	www.math.lsa.umich.edu /~gprasad/gpres.html (1190 words)

	Congruence Subgroups
		The genus of a subgroup U of Γ is the genus of the corresponding surface H/U. The principal congruence subgroup of level N, Γ(N), is the image in PSL(2,Z) of the group
		A subgroup of Γ which contains some principal congruence subgroup is called a congruence subgroup.
		We present complete tables of all congruence subgroups of PSL(2,Z) of genus 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24.
	www.math.tu-berlin.de /~pauli/congruence (942 words)

	[No title]
		In x6 we apply this to the level p congruence subgroups in Sp 4(Z), obtaining the cohomology of the relevant parabolic subgroups and from this a lower bound for the fourth betti number: Theorem.
		The application to the rational cohomology of congruence subgroups in SL3 (Z) is contained in the paper by Lee and Schwermer (see [LS]), although it is perha* *ps not as well known as it should be.
		SL 3(Z) induced by the inclusions of the parabolic subgroups.
	hopf.math.purdue.edu /Adem/amsbetti.txt (3065 words)

	Math 252: Congruence Subgroups
		This is the so-called "congruence subgroup problem." According to this MathSciNet review, if p is a prime, then every finite index subgroup of SL [1/p]) is a congruence subgroup, and for any n>2, all finite index subgroups of SL) are congruence subgroups.
		This paper by Hsu contains an algorithm to decide if a finite index subgroup is a congruence subgroup, and gives an example of a subgroup of index 12 that it is not a congruence subgroup.
		Let G be a finite index subgroup of SL Let R be a set of coset representatives for G. Define maps s and t from R to G as follows.
	modular.fas.harvard.edu /edu/Fall2003/252/lectures/09-19-03 (839 words)

	Structure of congruence subgroups (Site not responding. Last check: 2007-10-26)
		If G is a subgroup of finite index in PSL_2(Z), then returns a sequence of coset representatives of G in PSL_2(Z).
		For G a subgroup of PSL_2(Z) returns a sequence of points in the Upper Half plane which are the vertices of a fundamental domain for G. Example
		Returns a list of inequivalent elliptic points for the congruence subgroup G. A second argument may be given to specify the upper half plane H containing these elliptic points.
	www.math.niu.edu /help/math/magmahelp/text438.html (298 words)

	Congruence Subgroups of Groups Commensurable with PSL$(2,\Z)$ of Genus 0 and 1, C. J. Cummins
		Congruence Subgroups of Groups Commensurable with PSL$(2,\Z)$ of Genus 0 and 1, C. Cummins
		This result was used by the author and Pauli to compute the congruence subgroups of $\PSL(2,\Z)$ of genus less than or equal to 24.
		In this paper a result of Zograf is used to find a bound for the level of any congruence subgroup in terms of its genus.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1103749843 (315 words)

	Helena A. Verrill (Site not responding. Last check: 2007-10-26)
		Abstract: Congruence subgroups are certain subgroups of the group SL (Z) of integer 2 by 2 matrices with determinant 1.
		A Fundamental domain for a congruence subgroup is a region in the upper half plane which tells you about the quotient of the upper half plane by that group.
		If you take all the translates of a fundamental domain under the action of its congruence subgroup you get a tessellation of the upper half plane, similar to the pictures Escher drew on the disk.
	web.usna.navy.mil /~wdj/colloq/talk01_10.htm (195 words)

	Farey Symbols and Fundamental domains (Site not responding. Last check: 2007-10-26)
		One method of finding fundamental domains for congruence subgroups is the method of Farey Symbols, as described by Kulkarni [Kul91].
		Returns the generators of the congruence subgroup corresponding to the Farey symbol FS.
		Returns the coset representatives of the congruence subgroup of PSL_2(Z) corresponding to the Farey symbol FS.
	magma.maths.usyd.edu.au /magma/htmlhelp/text503.htm (514 words)

	Congruence Subgroups
		For congruence subgroups G and H, returns the index of G in H provided G is a subgroup of H.
		For G a congruence subgroup in PSL_2(Z), returns the index in PSL_2(Z).
		Returns the base ring over which matrices of the congruence subgroup G are defined.
	www.math.niu.edu /help/math/magmahelp/text437.html (521 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		There is a growing realization that there are deep connections between subgroup growth of a group and the algebraic structure of that group.
		A key element in the proof is the growth of congruence subgroups in arithmetic groups, a new kind of "non-commutative arithmetic", with applications to the study of lattices in Lie groups.
		An arithmetic group is a congruence group if the entries of each matrix in the group satisfy some elementary congruence conditions such as a particular entry is divisible by a fixed integer.
	www.math.psu.edu /oldColloquium/030501.html (203 words)

	Congruence subgroup - Definition up Erdmond.Com
		of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries.
		Further, we may take the inverse image of any subgroup (not just {e}) and get a congruence subgroup: the subgroups Γ0(N) important in modular form theory are defined in this way, from the subgroup of mod N 2x2 matrices with 1 on the diagonal and 0 below it.
		More generally, the notion of congruence subgroup can be defined for arithmetic_subgroups of algebraic_groups; that is, those for which we have a notion of 'integral structure' respected by the subgroup, and so some general idea of what 'congruence' means.
	www.erdmond.com /Congruence_subgroup.html (427 words)

	3. The Subgroup Problem (Site not responding. Last check: 2007-10-26)
		This section outlines the subgroup problem for groups and its connections to string rewriting techniques.
		Subgroups of groups can be characterized by one-sided congruences on the group.
		In the following we restrict ourselves to the case of right congruences (left congruences can be introduced in a similar fashion).
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/23/paper_html/node3.html (257 words)

	Jerusalem Mathematics Colloquium (Site not responding. Last check: 2007-10-26)
		This theory led to counting congruence subgroups in arithmetic groups.
		The latter counting is a kind of "non-commutative analytic number theory" where "counting primes" on one hand and delicate finite group theory, on the other hand, are combined.
		We will present the main counting results, applications to group theory and a connection with the congruence subgroup problem and the structure of the fundamental groups of hyperbolic manifolds.
	www.ma.huji.ac.il /~colloq/2001-02/col.011206.html (86 words)

	subgroup 1
		In mathematics, given a group G under an operation , we say that some subset H of G is a subgroup* if H is a group under * also.
		There is a minimal subgroup, the trivial group {e} (e being G's identity element), and a maximal subgroup, the group G itself.
		Given a subgroup H and some g in G, we define the left coset gH = {gh : h in H}.
	www.fact-library.com /subgroup_1.html (619 words)

	PlanetMath: multiplicative congruence
		if they are members of the same coset of the subgroup
		Cross-references: prime, modulus, multiplicative group, subgroup, coset, natural number, ring of integers, localization, finite prime, negative, positive, real numbers, real embedding, number field, real prime
		This is version 1 of multiplicative congruence, born on 2002-07-11.
	www.planetmath.org /encyclopedia/MultiplicativeCongruence.html (82 words)

	Jürgen Böhms Heimatseiten
		routines are made for computing fundamental domains of congruence subgroups of SL2(Z) in the upper half plane.
		They represent the congruence subgroup GAMMA which is its preimage in SL2(Z).
		They represent the congruence subgroup GAMMA which is their preimage in SL2(Z).
	www.aviduratas.de /compalg.html (775 words)

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