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Topic: Finitely generated

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	PlanetMath: finitely generated module
		"finitely generated module" is owned by Thomas Heye.
		Cross-references: contradiction, spanning set, polynomials, degrees, spanning, generated by, indeterminate, commutative ring, singleton, spanned by, cyclic, spans, subset, finite, ring, module
		This is version 9 of finitely generated module, born on 2003-10-15, modified 2006-06-17.
	planetmath.org /encyclopedia/FinitelyGeneratedRModule.html (111 words)

	PlanetMath: finitely generated group
		However, a finitely generated group may have subgroups that are not finitely generated.
		The finitely generated groups all of whose subgroups are also finitely generated are precisely the groups satisfying the maximal condition.
		This is version 20 of finitely generated group, born on 2002-02-03, modified 2006-04-11.
	planetmath.org /encyclopedia/FinitelyGenerated.html (192 words)

	Vector Enumeration
		The permutation module of degree 4 of this algebra is presented by 1 generator (as it is transitive) and the submodule generator b - 1.
		The generators and relators are as before, and now s=2 and the submodule generators are a^2 = {(b - 1, 0), (0, b - 1), (1 + a'(1 + a' + b), - 1 - a'(1 + a' + b))}.
		In example (3) the images of the two module generators are (1, 0, 0, 0, 0, 0, 0) and (0, 1, 0, 0, 0, 0, 0) while the basis vectors are images of (1, 0), (0, 1), (a', 0),(a, 0), (a^2, 0), (0, a') and (0, a).
	www.umich.edu /~gpcc/scs/magma/text939.htm (1542 words)

	[ref] 50 Finitely Presented Semigroups
		Finitely presented semigroups are obtained by factoring a free semigroup by a set of relations (a generating set for the congruence), ie, a set of pairs of words in the free semigroup.
		Finitely presented monoids are obtained by factoring a free monoid by a set of relations.
		If a finitely presented semigroup has a confluent rewriting system then it has a solvable word problem, that is, there is an algorithm to decide when two words in the free underlying semigroup represent the same element of the finitely presented semigroup.
	www.math.temple.edu /computing/gap/ref/CHAP050.htm (1545 words)

	[ref] 51 Finitely Presented Semigroups and Monoids
		Finitely presented monoids are obtained by factoring a free monoid by a set of relations, i.e.
		Calculations comparing elements of an finitely presented semigroup may run into problems: there are finitely presented semigroups for which no algorithm exists (it is known that no such algorithm can exist) that will tell for two arbitrary words in the generators whether the corresponding elements in the finitely presented semigroup are equal.
		The functionality available for finitely presented monoids is essentially the same as that available for finitely presented semigroups, and thus the previous sections apply (with the obvious changes) to finitely presented monoids.
	www.msri.org /about/computing/docs/gap/htm/ext/ref/CHAP051.htm (1798 words)

	Finitely-generated module - Wikipedia, the free encyclopedia
		A finite generating set would be a finite set of rational numbers which could, by raising them to any integer power and multiplying them together, be used to express any rational number.
		In the case where the module M is a vector space over a field R, and the generating set is linearly independent, n is well-defined and is referred to as the dimension of M (well-defined means that any linearly independent generating set has n elements: this is the dimension theorem for vector spaces).
		Finitely generated modules over the ring of integers Z coincide with the finitely generated abelian groups; these are completely classified.
	en.wikipedia.org /wiki/Finitely-generated_module (770 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		H is not finitely generated (every finitely generated subgroup of H is finite), but H is a subgroup of the group G generated by the two permutations (1,2) and (...,-3,-2,-1,0,1,2,3,...).
		More generally, if H is any countable group, then H can be embedded in a group G which is generated by two of its elements.
		H is not finitely generated (every finitely generated > subgroup of H is finite), but H is a subgroup of the group G > generated by the two permutations (1,2) and > (...,-3,-2,-1,0,1,2,3,...).
	www.math.niu.edu /~rusin/known-math/00_incoming/fin_gen (435 words)

	Groups with Word Problem in NP, and Higman Embeddings
		Nevertheless not every finitely presented group with polynomial Dehn function has a simply connected asymptotic cone because if the cone is symply connected then the group has a linear isodiametric function, and Theorem 3 allows one to construct lots of groups with polynomial Dehn functions which cannot have linear isodiametric functions.
		The class of finitely presented groups with polynomial Dehn functions is, by Theorem 14, the ``universal" subclass of the class of all groups with word problem in NP.
		Moreover, every finitely generated recursively presented group G can be embedded into a finitely presented group H in such a way that the degree of unsolvability of the conjugacy problem in H coinsides with the degree of undecidability of the conjugacy problem in G.
	www.math.vanderbilt.edu /~msapir/Talk1/node6.html (1879 words)

	The ascending tree condition
		From a constructive point of view, being finitely presented, rather than finitely generated, is a stronger property that must be assumed even if the ring is a field.
		For example, to prove that a quotient of a Noetherian module is Noetherian, you lift a chain of finitely generated ideals from the quotient to the module.
		If C is a set of finitely generated ideals with the property that a finitely generated ideal I is in C whenever all finitely generated ideals that strictly contain I are in C, then every finitely generated ideal is in C.
	www.math.fau.edu /Richman/Docs/new-acc.htm (2439 words)

	Wolfgang Hassler's homepage-Research interests
		This is equivalent to saying that H is a finitely generated Z-module, where Z denotes the ring of integers.
		In general it is still possible to decompose a finitely generated module over such a ring into a direct sum of indecomposable modules.
		For instance, the question which local rings admit finitely generated indecomposable modules having arbitrarily prescribed rank at the minimal primes was almost completely answered in the course of recent investigations.
	www.uni-graz.at /~hassler/research.htm (591 words)

	Classes of finitely generated convex sets of distributions (Site not responding. Last check: 2007-10-19)
		This sections demonstrates the generality of the previous algorithms by reducing a large number of classes of distributions to finitely generated credal sets.
		There are finitely many ways to distribute the mass among subsets, because x has finitely many values, so the convex set defined by the basic mass assignments is finitely generated.
		A reduction of the class to the class of finitely generated convex sets of distributions is still possible, since a density bounded class is a belief function class.
	cs.cmu.edu /~fgcozman/Research/QuasiBayesian/FiniteConvex/node10.html (1765 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.
		Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian.
		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 (489 words)

	[ref] 45 Finitely Presented Groups
		A finitely presented group (in short: FpGroup) is a group generated by a finite set of abstract generators subject to a finite set of relations that these generators satisfy.
		Finitely presented groups are obtained by factoring a free group by a set of relators.
		Finitely presented groups are groups and so all operations for groups should be applicable to them (though not necessarily efficient methods are available.) Most methods for finitely presented groups rely on coset enumeration.
	wwwmaths.anu.edu.au /research.programs/aat/GAP_manual/ref/CHAP045.htm (4867 words)

	Finitely generated abelian groups (Site not responding. Last check: 2007-10-19)
		We will now prove the structure theorem for finitely generated abelian groups, since it will be crucial for much of what we will do later.
		We first remark that any subgroup of a finitely generated free abelian group is finitely generated.
		Then we see that finitely generated abelian groups can be presented as quotients of finite rank free abelian groups, and such a presentation can be reinterpreted in terms of matrices over the integers.
	modular.fas.harvard.edu /edu/Spring2004/129/ant/html/node9.html (725 words)

	2. Residual Finiteness Results - Residually Finite Groups (Site not responding. Last check: 2007-10-19)
		The residual finiteness of free groups was again proven independently by Marshall Hall [MHall49], who in fact proved that free groups have the LERF property, a stronger property which will be discussed in section 4 of this paper.
		A large set of residually finite groups was revealed by Mal'cev's 1940 result [Mal40] that a finitely generated subgroup of a matrix group over a commutative ring with identity is residually finite.
		A and B are finitely generated torsion-free nilpotent and C is closed in A and B. A and B are finitely generated torsion-free nilpotent and C is cyclic (recall that Higman's example [Higm51] shows that one cannot weaken the conditions to A and B residually finite)
	www.math.umbc.edu /~campbell/CombGpThy/RF_Thesis/2_RF_Results.html (3347 words)

	Length Functions of a Finitely Generated Group
		In this section, we shall fix an arbitrary finitely generated group G and describe all possible length functions (and hence distortion functions) of G inside other groups.
		Although Theorem 9 shows that all ``reasonable" functions are length functions of a given finitely generated recursively presented group inside finitely presented groups, it does not give a characterization of these functions.
		By the proper choice of a universal group H it is not difficult to sharpen the formulation of Theorems 9 and 11.
	math.vanderbilt.edu /~msapir/Talk1/node5.html (924 words)

	Finitely Generated and Presented (Site not responding. Last check: 2007-10-19)
		Let the set s generate m, even if s is every element of m, and let f be a free module whose basis elements correspond to the generators in s.
		If the kernel of the homomorphism from f onto m is also finitely generated, then m is finitely presented.
		When a module is finitely generated, with n generators, endomorphisms are defined by n by n matrices over r that map each generator to a linear combination of said generators.
	www.mathreference.com /mod,fingen.html (381 words)

	[No title]
		However, we stress that the main results (Theorem 0.4 and Corollary 0.5 below) are true * for unsta- ble finitely* generated K - modules over an unstable noetherian algebra K (Theor* em 3.9 and Corollary 3.10) and the proof in the case of equivariant cohomology require s the same machinery as in the general* case.
		The finite generation part of Statement I is well known in case (a); for part (b) w* *e refer again to the appendix.
		Then the * class of unstable finitely* generated HBG - modules which are annihilated by some power of a (we will call such modules unstable a - torsion modules) forms a Serre class and we * will study localization away from the full subcategory T ors(a) of such modules.
	www.math.purdue.edu /research/atopology/Henn/kmod.txt (7623 words)

	Q not finitely generated
		I know that it helps to break it into steps, the first of which you show that any finitely generated subgroup of Q is contained in a cyclic subgroup (and hence is cyclic), and in the second step you show that Q itself is not cyclic.
		Hence Q is not cyclic and Q is not finitely generated.
		Q is a finitely generated field oveer Q, it is finitely generated as a module over its centre for instance.
	www.physicsforums.com /showthread.php?t=23106 (634 words)

	Finitely generated abelian group
		The finitely generated abelian groups, together with the group homomorphisms, form an abelian category.
		Expressing the theorem in general terms, it says a finitely-generated abelian group is the sum of a free abelian group and a finite abelian group, each of those being unique up to isomorphism.
		The converse isn't true however: there are many abelian groups of finite rank which are not finitely generated; the rank-1 group Q is one example, and the rank-0 group given by a direct sum of countably many copies of Z
	www.mik.fastload.org /fi/Finitely_generated_abelian_group.html (379 words)

	Magnus
		Every finitely generated abelian group can be decomposed into a direct product of finitely many infinite cyclic groups and finitely many cyclic groups of finite order.
		Since each such group is given by finitely many generators and finitely many defining relators in the category of abelian groups, i.e., as a quotient of a free abelian group of finite rank by a subgroup generated by finitely many elements, such a decomposition can be obtained by Gaussian Elimination.
		Each of the generators occuring in the given word is then re-expressed in terms of the new generators that have been obtained in the primary decomposition, leading to a representation of the given word in terms of the primary decomposition, i.e., its primary (canonical) form.
	zebra.sci.ccny.cuny.edu /web/caiss/MAGNUS/website/pages/checkinabeliangroup.html (1968 words)

	Magnus
		In general, the word problem for an amalgamated product is solvable when the factors are finitely presented groups with solvable word problem, if each of the amalagamated subgroups is finitely generated and if the generalized word problem for W in A and for Z in B is solvable.
		The integral homology groups of an amalgamated product G of two finitely presented groups A and B amalgamating the finitely generated subgroup H of A with the finitely generated subgroup K of B can be computed via the Mayer-Vietoris sequence (see the book by Hilton and Stammbach).
		Even though one knows that the second integral homology group of a finitely presented group is finitely generated, there is no algorithm which decides whether or not it is trivial (this is due to Cameron Gordon).
	zebra.sci.ccny.cuny.edu /web/caiss/MAGNUS/website/pages/checkinamalgamatedproduct.html (975 words)

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