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Topic: Finitely generated abelian groups

Related Topics

Abelian group

abelian categories

abelian varieties

free abelian group

finitely generated abelian groups

rank of an abelian group

abelian extensions

pre Abelian categories

category of abelian groups

Topological abelian group

arithmetic of abelian varieties

Abelian and tauberian theorems

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	Abelian group - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		In mathematics, an abelian group, also called a commutative group, is a group (G, ) such that a b = b * a for all a and b in G.
		Infinite-rank abelian groups can be extremely complex and many open questions exist, often intimately connected to questions of set theory.
		This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0.
	www.newlenox.us /project/wikipedia/index.php/Abelian_group (824 words)

	Algebraic topology (Site not responding. Last check: 2007-11-07)
		The fundamental group of a (finite) simplicial complex does have a finite presentation.
		Finitely generated abelian groups are completely classified and are particularly easy to work with.
		In general, all constructions of algebraic topology are functorial: the notions of category, functor and natural transformation originated here.
	www.sciencedaily.com /encyclopedia/algebraic_topology (479 words)

	Algebraic topology
		Finitely generated abelian group s can be completely classified and are particularly easy to work with.
		The free rank of the ''n -th homology group of a simplicial complex is equal to the n -th Betti number, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincaré characteristic.
		Modular representation theory of finite groups and related areas of algebraic topology; Homological algebra; Representation theory of finite-dimensional algebras.
	www.nebulasearch.com /encyclopedia/article/Algebraic_topology.html (835 words)

	Kids.net.au - Encyclopedia Algebraic topology -
		Fundamental groups give us crucial information about the structure of a topological space, but they are often nonabelian and can be difficult to work with.
		Finitely generated abelian groups can be completely classified and are particularly easy to work with.
		In particular, fundamental groups, homology and cohomology groups are invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups.
	www.kids.net.au /encyclopedia-wiki/al/Algebraic_topology (367 words)

	PlanetMath: abelian groups of order $120$ (Site not responding. Last check: 2007-11-07)
		Here we present an application of the fundamental theorem of finitely generated abelian groups.
		Since the group is finite it is obviously finitely generated, so we can apply the theorem.
		Notice that in the case of a finite group,
	planetmath.org /encyclopedia/AbelianGroupsOfOrder120.html (150 words)

	NTU Info Centre: Abelian category (Site not responding. Last check: 2007-11-07)
		The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups.
		Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A.
		Abelian categories were introduced by Alexander Grothendieck in the middle of the 1950s in order to unify various cohomology theories.
	www.nowtryus.com /article:Abelian_category (893 words)

	MA2161 Algebra II
		The aim of this course is to prove the theorem classifying the finitely-generated modules over a principal ideal domain, and to consider its applications to the classification of finitely-generated abelian groups and canonical forms of matrices.
		Essential for this course is the theory of groups and rings developed in MA2111, including Euclidean domains and their ideal structure, quotient rings and abelian groups.
		Group Concepts: Abelian group, cyclic group, generators of a group, finitely-generated group.
	www.mcs.le.ac.uk /Modules/MA/MA2161.html (611 words)

	Colloquium announcement for September 12, 2002 (Site not responding. Last check: 2007-11-07)
		Finite simple groups have been classified some twenty years ago and so it seems all what remains to do the same for finite groups is to find out how the simple groups "stack up".
		In case of finite groups of p-power order, where the only building block is the cyclic group of order p, "stacking up" is the real problem.
		In 1938, Reinhold Baer characterized the finitely generated abelian groups which are capable as those groups where the two factors of highest order in a direct composition have equal order.
	www.math.binghamton.edu /MATH/dept/colloquia/020912.html (281 words)

	BACKGROUND
		Note that the growth rate of the free metabelian group of rank 2 should be strictly less than 3 (which is the growth rate of the free group of rank 2) since a free metabelian group is not free.
		\ge 2, the matrices (1, a ; 0, 1) and (1, 0 ; a, 1) generate a free group.
		The automorphism group of a free metabelian group of finite rank is known to be finitely generated unless the rank equals 3 -- see [S.Bachmuth, H.Mochizuki, Aut(F) \to Aut(F/F") is surjective for free group F of rank \geq 4, Trans.
	www.cs.gc.cuny.edu /~cryptlab/gworld/problems/Back2.html (4773 words)

	Read This: Combinatorial Group Theory
		However, by the end of the century, finitely generated abelian groups had been classified, and von Dyck had introduced free groups and presentations of groups by generators and relations.
		Thus in the first chapter the group defined by a presentation is constructed as a group of equivalence classes of words.
		A free group is then defined as one given by a presentation whose set of relators is empty, and the definition by the universal property is relegated to an exercise.
	www.maa.org /reviews/MagnusKarrassSolitar.html (1163 words)

	Re: The fundamental group of a topological group (Site not responding. Last check: 2007-11-07)
		But since some physicists might have skipped that course, let me say what the result is: a finitely generated abelian group is just a product of finitely many copies of Z (the integers) together with a finite collection of groups of the form Z/n (the integers mod n).
		So, the really fun question is "which finitely generated abelian groups arise as the fundamental group of some compact Lie group?" The answer is not hard, but let me explain how I blundered my way to it, because it might be interesting to people who want to learn stuff about Lie groups.
		This group N may not split up neatly as N1 x N2 where N1 is a normal subgroup of G and N2 is a normal subgroup of T. However, it's pretty easy to see that N lies inside a finite normal subgroup of the form N1 x N2.
	www.lns.cornell.edu /spr/2002-02/msg0039540.html (892 words)

	Graduate Math Courses
		Algebraic topics are chosen from Lie groups and Lie algebras, non-associative algebras, abelian groups, classical groups, algebraic number theory, representation theory, algebraic geometry, and ring theory.
		In addition to these general requirements, It is also requested that the graduate student has a previous knowledge of a programming language.
		Fundamental groups and the Van Kampen theorem; simplicial complexes, simplicial homology, and Euler characteristic; singular homology, Mayer-Vietoris sequences.
	www.gwu.edu /~math/graduate/gradCourses.html (1077 words)

	Graduate Mathematics Courses (Site not responding. Last check: 2007-11-07)
		Groups, subgroups, cyclic groups, quotient groups, Lagranges Theorem, permutation groups, homomorphism and isomorphism theorems, Cayley's theorem, rings, subrings, ideals, fields, homomorphism and isomorphism theorems.
		Sylow's theorems for finite groups, p-groups, abelian groups, group action on sets, domains, prime and maximal ideals, unique factorization domain.
		Groups, group actions on sets, structure of finitely generated abelian groups, category theory, exact sequences, rings, P.l.D's, modules, projective, injective and free modules.
	donaldson.math.howard.edu /~reb/gradcour.htm (1023 words)

	PlanetMath: (Site not responding. Last check: 2007-11-07)
		finite intersection condition (=finite intersection property) owned by bbukh
		finite projective module (=finitely generated projective module) owned by mhale
		finite ring has no proper overrings owned by pahio
	planetmath.org /encyclopedia/F (1695 words)

	Revisit the first problem set
		Definition 1.2 A group is finitely generated if it has a finite set of generators.
		The most interesting cases would be classes of simple groups, starting with the alternating groups and the projective special linear groups.
		It would be of interest to prove this various inifinite classes of finite simple groups, such as the projective special linear groups.
	people.cs.uchicago.edu /~laci/reu03/n2_6/node1.html (208 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Subject: Re: abelian groups Date: Mon, 19 Mar 2001 14:15:02 +0100 Newsgroups: sci.math Summary: Finitely generated abelian groups satisfy ascending chain condition.
		Let 0 -> A' -> A -> A'' -> 0 be a short exact sequence of abelian groups.
		abelian group is a quotient of a f.g.
	www.math.niu.edu /~rusin/known-math/01_incoming/ACC (142 words)

	Publications
		Goldstein, R.Z. and Turner, E.C., Monomorphisms of free groups have finitely generated centralizers, Invent.
		Collins, D.J. and Turner, E.C. An automorphism of a free group of finite rank with maximal rank fixed point subgroup fixes a primitive element, J. Pure and Applied Algebra 88 (1993), 43--49.
		Turner, E.C. and Voce, D.A., Tame automorphisms of finitely generated abelian groups, Proceedings of the Edinburgh Math.
	math.albany.edu:8000 /~ted/pub.html (488 words)

	Ergodic Theory
		During the summer of 2001 I was advising a group of four students in the investigation of Root Groups.
		They are composed of groups familiar to any undergraduate, the integers and finite cyclic groups, yet pieced together with a simple and unusual operation.
		We also generalized the definition of root group to include groups where the carry could be a natural number m different from 1.
	www2.potsdam.edu /madorebf/reu2001.htm (1501 words)

	Square-free - Wikipedia, the free encyclopedia
		The positive integer n is square-free if and only if all abelian groups of order n are isomorphic, which is the case if and only if all of them are cyclic.
		This follows from the classification of finitely generated abelian groups.
		The integer n is square-free iff the factor ring Z / nZ (see modular arithmetic) is a product of fields.
	en.wikipedia.org /wiki/Square-free (381 words)

	[No title]
		The free rank of the n-th homology group of a simplicial complex is equal to the n-th Betti number, so one can use the homology groups of a simplicial complex to calculate its
		Fundamental groups, homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are
		homeomorphic have the same associated groups; a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings.
	en-cyclopedia.com /wiki/Algebraic_topology (437 words)

	Generic Abelian Groups (New) [HB 25] (Site not responding. Last check: 2007-11-07)
		This is possible if the order of the group is known beforehand.
		However, it is also possible to compute the structure of the group from a user-supplied set of generators.
		It is also possible to compute the representation of any group element in terms of a given set of generators of the group.
	magma.maths.usyd.edu.au /magma/ReleaseNotes/rel28/node13.html (114 words)

	Amazon.com: Books: Presentations of Groups (London Mathematical Society Student Texts) (Site not responding. Last check: 2007-11-07)
		W Bruce (Series Editor) "The fundamental notion underlying the theory of group presentations is that of a free group..." (more)
		The fundamental notion underlying the theory of group presentations is that of a free group.
		relator tables, metacyclic groups, definition bonus, defining relators, dimension subgroups, augmentation ideal, finite presentation, boundary label, free generators, normal closure, central series, invariant factors, permutation representation, free abelian group, monitor table, infinite cyclic group, elementary abelian, substitution test, normal subgroup, short exact sequence, reduced words, trivial group, basis theorem, relation matrix, finite index
	www.amazon.com /exec/obidos/tg/detail/-/0521585422?v=glance (542 words)

	MT381 Algebra III (Term 2: Dr Y Barnea)
		The aim of this course is to give the students a working knowledge of finitely-generated abelian groups and the analogous theory of modules.
		demonstrate the existence of normal subgroups of certain finite groups, and to use this to determine the isomorphism classes of groups of certain orders;
		Group Theory: Homomorphisms, normal subgroups, factor groups, the first isomorphism theorem.
	www.rhul.ac.uk /mathematics/bluebook/courses/MT381.htm (222 words)

	ScholarlyActivities
		On the Generalization of the Climbing Stairs Problem, presented at the fall meeting of the Indiana Section of the Mathematical Association of America, Goshen College, IN, October 2003.
		On the Climbing Stairs Problem: a Generalization, presented at the spring meeting of the Missouri Section of the MAA, Washington University, St. Louis, MO, April 2003.
		On the Generalized Free Product of Finitely Generated Nilpotent Groups and the Near Frattini Subgroup, presented at the joint annual meeting of the American Mathematical Society and Mathematical Association of America, Baltimore, MD, Jan. 1998.
	faculty.evansville.edu /ma3/scholarlyactivities.html (1672 words)

	Atlas: Lattice-ordered Abelian groups and Toric varieties by Vincenzo Marra (Site not responding. Last check: 2007-11-07)
		This talk is an attempt to explain how one relates questions about lattice-ordered Abelian groups to questions in algebraic geometry.
		In a parallel and hitherto independent development whose full import is just beginning to be appreciated, K. Baker and W. Beynon brought to light in the 1970s an intimate connection between finitely generated projective Abelian l-groups and supports of rational polyhedral sets.
		We show that a finitely generated projective Abelian l-group with a distinguished generating set of elements satisfying a certain condition completely encodes the absolute geometry (i.e.
	atlas-conferences.com /cgi-bin/abstract/cakg-16 (316 words)

	Discrete Mathematics Graduate Courses (Site not responding. Last check: 2007-11-07)
		Review of elementary group theory, permutation groups, group actions, Sylow theorems, semi-direct and wreath products, classical matrix groups, automorphism groups of combinatorial structures, Polya enumeration.
		Further topics may include representation theory, free groups, finite groups of isometries of Euclidean spaces.
		Topics vary from year to year and will be chosen from a variety of areas in pure mathematics (not necessarily discrete mathematics).
	www.math.udel.edu /research/DiscreteMath/courses.html (241 words)

	[No title]
		It assumes that the generators ) ( are symbols, and will not work if they are linear combinations of ) ( symbols.
		I think it is also ", "obtained by applying the unstable J-homomorphism to the generator of ", "\\(\\pi_3SO(3)\\).
		We hope to be ) ( able to write x in terms of standard generators, at which point this ) ( information will become redundant.
	www.shef.ac.uk /nps/toda/Toda.m (2600 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Ordered Generating System for finite non-abelian groups are introduced.
		This concept generalizes the basis theorem for finitely generated abelian groups.
		Interesting examples include symmetric groups and other Coxeter groups, alternating groups and matrix groups over a finite field.
	www.math.technion.ac.il /~techm/20041128141520041128sch (145 words)

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