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Topic: Fundamental theorem of finitely generated abelian groups

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	PlanetMath: fundamental theorem of finitely generated abelian groups
		"fundamental theorem of finitely generated abelian groups" is owned by alozano.
		This is version 2 of fundamental theorem of finitely generated abelian groups, born on 2003-08-25, modified 2004-03-17.
		(Group theory and generalizations :: Structure and classification of infinite or finite groups :: General structure theorems)
	planetmath.org /encyclopedia/FundamentalTheoremOfFinitelyGeneratedAbelianGroups.html (80 words)

	Finitely generated abelian group - Wikipedia, the free encyclopedia
		The fundamental theorem of finitely generated abelian groups states that every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups.
		Stated differently the fundamental theorem says that a finitely-generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism.
		The finitely generated abelian groups, together with the group homomorphisms, form an abelian category which is a Serre subcategory of the category of abelian groups.
	en.wikipedia.org /wiki/Finitely_generated_abelian_group (483 words)

	Abelian group - Wikipedia, the free encyclopedia
		Abelian groups are named after Niels Henrik Abel.
		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.
		Given this, the fundamental theorem shows that to compute the automorphism group of G it suffices to compute the automorphism groups of the Sylow p-subgroups separately (that is, all cyclic subgroups with order a power of p).
	en.wikipedia.org /wiki/Abelian_group (1264 words)

	PlanetMath: abelian groups of order $120$
		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)

	Abelian group (Site not responding. Last check: 2007-11-03)
		In mathematics, an abelian group is a commutative group, i.e.
		There are two main notational conventions for abelian groups -- additive and multiplicative.
		Many large abelian groups carry a natural topology, turning them into topological groups.
	www.gogoglo.com /wiki/en/wikipedia/a/ab/abelian_group.html (791 words)

	Graduate Study in Algebra
		The "symmetric" groups of all permutations of a set are investigated, and the Cayley theorem (showing an arbitrary abstract group may be regarded as a subgroup of some symmetric group) is proved.
		The goal of the course is the fundamental theorem of Galois theory and the solutions to the three pearls of antiquity: the quadrature of the circle, the trisection of an angle, and the duplication of the cube.
		The following topics are studied: the isomorphism theorems for groups, solvability of p-groups, simplicity of the alternating group on at least 5 letters, Sylow theorems, Jordan-Holder Theorem, principal ideal domains, Gauss' lemma, Eisenstein's criterion, the fundamental theorem of Galois theory, finite fields, cyclotomic fields, solvability of equations by radicals.
	www.math.uiuc.edu /GraduateProgram/researchmath/gradalgebra.html (1660 words)

	Mathenomicon.net : abelian (Site not responding. Last check: 2007-11-03)
		Examples of abelian groups include the integers (under addition), the cyclic groups, direct products of cyclic groups, and the complex numbers excluding zero (under multiplication).
		The smallest non-abelian group is the symmetric group on three elements.
		Finitely generated abelian groups are characterized by the fundamental theorem of same.
	www.cenius.net /refer/articles/a/abelian/abelian.html (105 words)

	MTH-3D15 : Theory of Finite Groups
		Group theory is a large topic which interconnects with many branches of pure and applied mathematics.
		Abstract groups began to emerge with Jordan's seminal Traité des substitutions et des equations algébriques (1870) while the definition of abstract groups in general appears to be due to Weber (1882).
		For finite groups the orbit stabilizer theorem, a relatively easy result on group actions, plays a central role and many theorems appear as a consequence of it.
	www.mth.uea.ac.uk /maths/syllabuses/0506/4D1505.html (709 words)

	Senior Paper (Site not responding. Last check: 2007-11-03)
		Subgroup- A subset of a group that is a group under the same binary operator is a subgroup.
		Note- The cyclic subgroup generated by a is a subgroup of the group a is in.
		Group Isomorphism- this means that there is a (well-defined) function from G onto H that is one to one and a homomorphism.
	www.public.iastate.edu /~zicktimo/Undergrad/seniorpaper.htm (3003 words)

	[No title]
		Determine the structure of the commutative group defined by generators a,b,c and relations 3a + 9b + 9c = 0 and 9a - 3b + 9c = 0.
		Given the generators and relations 3a + 6 b + 12 c = 0 8a + 2 b + 6 c = 0 2a + 3 b + 15 c = 0 Decompose the group into cyclic summands and give the generators of each cyclic summand.
		Given the generators and relations 8a + 5 b + 12 c = 0 6a + 3 b + 2 c = 0 6a + 9 b + 6 c = 0 Decompose the group into cyclic summands and give the generators of each cyclic summand.
	orion.math.iastate.edu /hentzel/class.301.03/Oct.01 (861 words)

	Application to finitely generated Abelian groups
		Unfortunately, customarily, the concepts of orders in Group Theory are in conflict with the definitions we provided for the order of any element in a module and the order of a torsion module itself.
		, is defined to be the cardinality of the group
		With the convention stated in the last paragraph, as a corollary of the last theorem, we have the following important structural theorem for finitely generated abelian groups:
	amath.nchu.edu.tw /~hsu/run/node75.html (302 words)

	Spring 1995 Algebra Prelim Solutions
		In particular G is abelian and by the fundamental structure theorem for finitely generated abelian groups, these last two groups are not isomorphic.
		Let G be a finitely generated abelian group with the given property.
		We conclude that the finitely generated abelian groups with the property that for all subgroups A, B, either
	www.math.vt.edu /people/linnell/Teaching/Algprelims/Spring95sol (698 words)

	Abelian group (Site not responding. Last check: 2007-11-03)
		It frees woman from the common diseases frequently suffered such as delayed and irregular menstruations, back-aches and stomach-aches along menstruation and even to tighten stomach muscles and uterus muscles.
		Every field gives rise to two abelian groups in the same fashion -- the additive group of all elements, and the multiplicative group of nonzero elements.
		The abelian groups, together with group homomorphisms, form a category, the prototype of an abelian category.
	www.aseannewsnetwork.de /articles/content/a/ab/abelian_group.html (821 words)

	Van Wyk's 431 Sections
		quotient groups), simple groups, maximal normal subgroups, the center of a group, the commutator subgroup of a group, why the converse of Lagrange's Theorem is false.
		Groups with bases (known as free abelian groups), why each is isomorphic to a direct product of a bunch of copies of the integers, and a fairly technical proof of the Fundamental Theorem of Finitely Generated Abelian Groups (don't get too hung up on this part of the section).
		Reduced words, free group F[A] generated by a set A, rank, homorphisms of free groups, the fact that every group is a homomorphic image of a free group.
	www.math.jmu.edu /~vanwyk/courses/431/hw.html (864 words)

	Homework for Math 120
		You should read 5.1 and understand the statement of the fundamental theorem in 5.2 (I hope we'll have time for the proof before the end of the quarter).
		Let G be the group whose elements are pairs [x,y] of real numbers with x not 0, and whose operation is [a,b][c,d] = [ac,ad+b].
		More generally, in a deck of 2n cards, the shuffle is the permutation of the nonzero numbers mod (2n+1) by the map f(j)=2j.
	math.stanford.edu /~white/120/120hw8.htm (751 words)

	Topology II Problem List, page 4 (Site not responding. Last check: 2007-11-03)
		Then show that every finitely generated abelian group is the fundamental group of a path connected topological space.
		Show that the fundamental group of a connected graph with Euler characteristic e is a free group of rank 1-e.
		The fundamental group of X is a free group F(a,b) of rank 2.
	www.math.ou.edu /~amiller/topology/page4.htm (495 words)

	PlanetMath 2004-01-12 Snapshot: F
		finite extension of fields is an algebraic extension, a
		fundamental theorem of calculus (=fundamental theorems of calculus)
		fundamental theorem of finitely generated abelian groups (defined in fundamental theorem of finitely generated abelian groups)
	simba.cs.uct.ac.za /~hussein/PlanetMath-snapshot_2004-01-12/F.html (193 words)

	Ergodic Theory
		During the summer of 2001 I was advising a group of four students in the investigation of Root Groups.
		We also generalized the definition of root group to include groups where the carry could be a natural number m different from 1.
		Using this theorem it is easy to show that root groups are a very specific class of finitely generated Abelian groups.
	www2.potsdam.edu /madorebf/reu2001.htm (1494 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		This is false, and one of the groups could be abelian.
		Otherwise isomorphism testing would be much easier than it is! Groups of prime power order are often a good place to look for counterexamples, and a brute force computer search is as good a way as any to find them.
		Monthly 72 (1965), 48--50; MR 30 #1176] proved that a finite abelian group is determined up to isomorphism by its order and the orders of each of its elements.
	www.math.niu.edu /~rusin/known-math/99/same_orders (349 words)

	Math 120 Final Exam Information (Site not responding. Last check: 2007-11-03)
		Be able to use the class equation, Sylow's theorem(s), and the fundamental theorem on finitely generated abelian groups.
		Suppose G is a group of order 90 with subgroups A of order 6 and B of order 10.
		Prove also that every finitely generated ideal in R is a principal ideal.
	math.stanford.edu /~white/120/fininfo.htm (257 words)

	Tulane Math Graduate Algebra Qualifying exam syllabus
		Symmetric and alternating groups, permutation groups, Cayley's theorem.
		Direct sums, fundamental theorem* on finitely generated abelian groups.
		Galois groups, solvability of algebraic equations, constructions with straight edge and compass.
	www.math.tulane.edu /graduate/qualifying/algebra.html (148 words)

	Algebraic Groups
		Theorem 21: The intersection of any collection of subgroups H
		is finite and generates G, then we say that G is finitely generated.
		Theorem 23 (Fundamental Theorem of Finitely Generated Abelian Groups): Every finitely generated abelian group G is isomorphic to a direct product of cyclic groups of the form
	www1.hollins.edu /faculty/clarkjm/AlgebraicGroups/class9.htm (293 words)

	MAT 444 (Site not responding. Last check: 2007-11-03)
		On page 14 (2nd version), in the Theorem on commutator subgroups, in the first part I forgot to say the homomorphism from G to H should be assumed onto.
		The coverage on the final exam is the entire course, although the material covered after the midterm has higher priority for inclusion in the final exam.
		In general, the topics appearing on the midterm and the final exam are a sampling of the important material from the course, and not every topic covered in the semester can appear on the midterm or the final exam.
	math.la.asu.edu /~quigg/teach/courses/444/2001s/coursemain.html (1173 words)

	The Math Major Vol. 2, No. 5 (Site not responding. Last check: 2007-11-03)
		Students are required to participate in a group project.
		291: (Seminar in "Group Theory"--Dr. Sun) The content is mostly finite groups, a bit of free groups and infinite abelian groups.
		Finite groups include the theorems of Sylow, Cauchy, Cayley, Frobenious, Jordan-Holder-Schreier and Burnside, p-groups, solvable groups, nilpotent groups, simple groups, permutation groups, linear groups, etc. Infinite abelian groups include the Fundamental Theorem of Finitely Generated Abelian Groups.
	www.csufresno.edu /math/department/newsletter/vol2/v2n05.html (854 words)

	New Page 2 (Site not responding. Last check: 2007-11-03)
		Let G be a group of order pq, where p and q are distinct primes.
		Let G be a group and Z(G) the center of G.
		Any group of prime order is cyclic, and the trivial group is cyclic.
	cas.memphis.edu /rfaudree/E24261.htm (461 words)

	Syllabus Math 200B
		Fundamental theorem for finitely generated modules over a PID(invariant factor form) 12.1, pp.438-443.
		The fundamental theorem for finitely generated abelian groups, 5.2.
		The fundamental theorem of algebra and the first fundamental theorem for S
	math.ucsd.edu /~nwallach/Syll200B.html (43 words)

	Spring 1980 Algebra Prelim Solutions
		We shall use the fundamental theorem for finitely generated abelian groups.
		Suppose by way of contradiction G is a simple group of order 56.
		Sylow's theorem for the prime 7 shows that the number of Sylow 7-subgroups is congruent to 1 modulo 7 and divides 8, hence the number of Sylow 7-subgroups is 1 or 8.
	www.math.vt.edu /people/linnell/Teaching/Algprelims/Spring80sol (625 words)

	Teaching page of Alexander Yong (Site not responding. Last check: 2007-11-03)
		The integers, congruences and the Fundamental Theorem of Arithmetic.
		Cyclic groups, any subgroup of a cyclic group is cyclic (idea of proof: in the end, consider the division algorithm).
		Lecture 4: Cayley's theorem (idea of proof: any row of the group table of G is a permutation of G), introduction to group actions: orbits, cycles of permutations.
	math.berkeley.edu /~ayong/teaching_Math113_Fall2003.html (2733 words)

	MATH 335 -- Modern Algebra -- Spring 2006
		The abstract point of view, based on an axiomatic approach, reveals many deep ideas behind seemingly innocent structures--such as the arithmetic of counting numbers--and serves as an elegant organizing tool for the vast universe of modern algebra.
		Generations of brilliant minds have crystallized these ideas in the ideas in the concept of groups, rings, fields, modules, and their quotient structures and homomorphisms--the topics of MATH 335 & 435.
		We will not strive for the maximal possible generality but rather work out as many concrete examples/incarnations of theoretical concepts as possible.
	math.sfsu.edu /beck/teach/335.html (517 words)

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