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Topic: Nilpotent

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	PlanetMath: nilpotent group
		The nilpotency class or nilpotent class of a nilpotent group is the length of the lower central series (equivalently, the length of the upper central series).
		Nilpotent groups are related to nilpotent Lie algebras in that a Lie group is nilpotent as a group if and only if its corresponding Lie algebra is nilpotent.
		This is version 5 of nilpotent group, born on 2002-06-16, modified 2006-11-01.
	planetmath.org /encyclopedia/Nilpotent5.html (189 words)

	Nilpotent group - Wikipedia, the free encyclopedia
		Nilpotent groups arise in Galois theory, as well as in the classification of groups.
		Every abelian group is nilpotent of class 1, except for the trivial group, which is nilpotent of class 0.
		Every subgroup of a nilpotent group of class n is nilpotent of class at most n; in addition, if f is a homomorphism of a nilpotent group of class n, then the image of f is nilpotent of class at most n.
	en.wikipedia.org /wiki/Nilpotent_group (597 words)

	PlanetMath: nilpotent matrix
		Similarly, since the trace of a square matrix is the sum of the eigenvalues, it follows that it has trace 0.
		One class of nilpotent matrices are the strictly triangular matrices (lower or upper), this follows from the fact that the eigenvalues of a triangular matrix are the diagonal elements, and thus are all zero in the case of strictly triangular matrices.
		This is version 14 of nilpotent matrix, born on 2002-10-16, modified 2006-06-13.
	planetmath.org /encyclopedia/NilpotentMatrix.html (184 words)

	NILPOTENT GROUPS (Site not responding. Last check: 2007-11-02)
		(N1) (A.Miasnikov) Let G be a free nilpotent group of finite rank.
		(N2) Let G be a finitely generated nilpotent group.
		(N4) (G.Baumslag) Let G be a finitely generated torsion-free nilpotent group.
	zebra.sci.ccny.cuny.edu /web/nygtc/problems/probnil.html (168 words)

	Nilpotent - Term Explanation on IndexSuche.Com (Site not responding. Last check: 2007-11-02)
		In the factor_ring Z/9Z, the class of 3 is nilpotent because 32 is congruent to 0 modulo 9.
		The nilpotent elements from a commutative_ring form an ideal; this is a consequence of the binomial_theorem.
		Every nilpotent element in a commutative ring is contained in every prime_ideal of that ring, and in fact the intersection of all these prime_ideals is equal to the nilradical.
	www.indexsuche.com /Nilpotent.html (261 words)

	Nilpotent (Site not responding. Last check: 2007-11-02)
		No nilpotent element can be a unit (except in thetrivial ring {0} which has only a single element 0=1).
		The nilpotent elements from a commutative ring form an ideal ; this is a consequence of the binomial theorem.
		Every nilpotent element in a commutative ring is contained in every primeideal of that ring, and in fact the intersection of all these prime ideals is equal to the nilradical.
	www.therfcc.org /nilpotent-210832.html (194 words)

	Nilpotent group (Site not responding. Last check: 2007-11-02)
		For a justification of the term nilpotent start with a nilpotent group G an element g of G and define a function f : G → G by f (x) = [ x g ].
		Every subgroup of a nilpotent group of n is nilpotent of class at most n ; in addition if f is a homomorphism of a nilpotent group of class n then the image of f is nilpotent of class at most n.
		It is supposed to develop the general theory of nilpotent Lie algebras, but the most part of the book deals with a quite special class of nilpotent algebras, called filiform.
	www.freeglossary.com /Nilpotent_group (715 words)

	Nilpotent - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-02)
		In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that x
		No nilpotent element can be a unit (except in the trivial ring {0} which has only a single element 0 = 1).
		Every nilpotent element in a commutative ring is contained in every prime ideal of that ring, and in fact the intersection of all these prime ideals is equal to the nilradical.
	encyclopedia.worldsearch.com /nilpotent.htm (301 words)

	Nilpotent - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-02)
		It can be shown that an n×n matrix A is nilpotent if and only if its characteristic polynomial is given by
		An operator $Q that satisfies Q^2=0 is$ nilpotent.
		The BRST charge is an important example in physics.
	www.encyclopedia-online.info /Nilpotent (235 words)

	Re: Nilpotent fuzz (Site not responding. Last check: 2007-11-02)
		"nilpotent fuzz" is originally an idea from "ordinary" geometry (as opposed to "super" geometry).
		it turned out that nilpotence _did_ have a geometric interpretation after all, and that the geometric interpretation of nilpotence had secretly been a central theme (perhaps even _the_ central theme) of mathematics for hundreds of years.
		nilpotence is a comic exaggeration of the real properties of real small nonzero numbers.
	www.lns.cornell.edu /spr/2000-10/msg0028834.html (777 words)

	Computer search for nilpotent complexes, Robert H. Lewis, Guy D. Moore
		The concept of nilpotency for a topological space is a generalization of simple connectivity.
		In 1985 Lewis showed that for any finite nilpotent group there is a (not necessarily finite) three-dimensional nilpotent complex with that fundamental group.
		The smallest finite nilpotent group for which it was unknown whether a finite three-dimensional nilpotent complex exists was $\bold Z_2 \oplus \bold Z_6$.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1047920423 (253 words)

	Nilpotent group - Term Explanation on IndexSuche.Com
		Then this function is nilpotent in the sense that there exists a natural_number ''n'' such that ''f''''n'', the ''n''-th iteration of ''f'', sends every element ''x'' of ''G'' to the identity element.
		Since each successive factor_group ''Z'i''+1/''Z'i'' is abelian, and the series is finite, every nilpotent group is a solvable_group with a relatively simple structure.
		The last statement can be extended to infinite groups: If ''G'' is a nilpotent group, then every Sylow subgroup ''G''''p'' of ''G'' is normal, and the direct sum of these Sylow subgroups is the subgroup of all elements of finite order in ''G'' (see torsion_subgroup).
	www.indexsuche.com /Nilpotent_group.html (579 words)

	Nilpotent groups (Site not responding. Last check: 2007-11-02)
		Our odyssey begins with nilpotent groups, which are in many ways the second simplest kind of groups there are (the simplest being abelian groups).
		The nilpotency class of G is, in that case, the smallest n for which G(n) is trivial.
		The class of all nilpotent groups (of any class) is not, however, a variety, since it is not closed under arbitrary direct products; it is only a quasivariety.
	math.berkeley.edu /~magidin/research/nils.html (241 words)

	Nilpotent group (Site not responding. Last check: 2007-11-02)
		For a justification of the term nilpotent, start with a nilpotent group G, an element g ofG and define a function f : G → G by f(x) =[x,g].
		The discrete Heisenberg group is anotherexample of non-abelian nilpotent group.
		Every subgroup of a nilpotent group of class n is nilpotent of class at most n; in addition, if fis a homomorphism of a nilpotent group of class n,then the image of f is nilpotent of class at most n.
	www.therfcc.org /nilpotent-group-210807.html (540 words)

	Citations: Nilpotent structures and invariant metrics on collapsed manifolds - Cheeger, Fukaya, Gromov (ResearchIndex)
		Citations: Nilpotent structures and invariant metrics on collapsed manifolds - Cheeger, Fukaya, Gromov (ResearchIndex)
		Cheeger, K. Fukaya and M. Gromov, Nilpotent Structures and Invariant Metrics on Collapsed Manifolds, J. Amer.
		Cheeger; K. Fukaya; M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Splittings And...-Structures For Manifolds With..
	citeseer.ist.psu.edu /context/502153/0 (1212 words)

	Combing nilpotent and polycyclic groups (Site not responding. Last check: 2007-11-02)
		The notable exclusions from the family of automatic groups are those nilpotent groups which are not virtually abelian, and the fundamental groups of compact $3$-manifolds based on the $Nil$ or $Sol$ geometries.
		It is shown that a finitely generated class 2 nilpotent group with cyclic commutator subgroup is real-time combable, as are also all 2 or 3-generated class 2 nilpotent groups, and groups in specific families of nilpotent groups (the finitely generated Heisenberg groups, groups of unipotent matrices over $\Z$ and the free class 2 nilpotent groups).
		All the combings constructed in the article are boundedly asynchronous, and those for nilpotent-by-finite groups have polynomially bounded length functions, of degree equal to the nilpotency class, $c$; this verifies a polynomial upper bound on the Dehn functions of those groups of degree $c$+1.
	www.mas.ncl.ac.uk /~nser/abstracts/nilpotent.html (271 words)

	[No title]
		The theory was organized around a familyof "higher periodicities" generalizing Bott periodicity, and dependedon being able to determine the nilpotent and non-nilpotent maps in the category of spectra.
		R; from the sphere spectrum to aring spectrum, is nilpotent if it is nilpotent when regarded as an element of the ring ss R. The main result of [7] is Theorem 2.
		Theorem 7 is in fact equivalent to the Nilpotence Theorem (the proof is sketched atthe end of Section 4).
	hopf.math.purdue.edu /Hopkins-SmithJH/nilpII.oneside.abstract (755 words)

	Nilpotent (Site not responding. Last check: 2007-11-02)
		Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis.
		Both are linked, also through supersymmetry and Morse theory, as shown by Edward Witten in a celebrated article.
		The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.
	nba.servegame.org /en/Nilpotent.htm (339 words)

	Abelian, Nilpotent and Soluble Quotient
		A nilpotent quotient algorithm constructs, from a finite presentation of a group, a polycyclic presentation of a nilpotent quotient of the finitely presented group.
		The nilpotent quotient algorithm used by Magma is the Australian National University Nilpotent Quotient program, as described in [Nic96].
		G has infinite nilpotent quotients if and only if G/G_1 (the maximal abelian quotient of G) is infinite and a prime p divides a finite cyclic factor of a nilpotent quotient if and only if p divides a cyclic factor of G/G_1.
	www.umich.edu /~gpcc/scs/magma/text297.htm (2923 words)

	Publications
		An algorithm for computing nilpotent approximations for invariant systems on Lie groups is presented.
		This paper develops a method for constructing nilpotent approximations for local representations of invariant systems on matrix Lie groups via a simple operation on the structure constants of the associated Lie algebra.
		The crucial role such nilpotent approximations play for the problem of feedback nilpotentization is discussed.
	www.cds.caltech.edu /~herb/publications.html (683 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		I know what a nilpotent semigroup is: the "opposite" >of a group, namely a semigroup S with one idempotent '0', generated >by each x in S, with 0 as final iteration.
		But then the question is why nilpotent Lie algebras are so called.
		Engel's theorem represents such an algebra by a Lie algebra of matrices, and each of these matrices is nilpotent in the sense that some power is 0.
	www.math.niu.edu /~rusin/known-math/99/nilpotent_gp (234 words)

	Lie Algebra Notes
		Theorem: Suppose L is a Lie subalgebra of gl(V) consisting entirely of nilpotent endomorphisms, then there is some v in V so that xv=0 for all x in L. Proof: We proceed by induction on dim L. The statement is obviously true for dim L in {0,1}.
		So yw is in W for w in W. As y maps W to W and y is a nilpotent endomorphism of V, y is a nilpotent endomorphism of W. Thus there exists non-zero v in W so yv=0.
		Corollary: Suppose L is a nilpotent subalgebra of gl(V).
	www.math.rutgers.edu /~nacin/Sahi4.html (1540 words)

	CiteULike: Nilpotent Classical Mechanics (Site not responding. Last check: 2007-11-02)
		The formalism of nilpotent mechanics is introduced in the Lagrangian and Hamiltonian form.
		As an example of an η-system the nilpotent oscillator is introduced and its supersymmetrization considered.
		As an example of an \eta-system the nilpotent oscillator is introduced and its supersymmetrization considered.
	www.citeulike.org /user/ansobol/article/842069 (237 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		G is said to have nilpotency class c if c is the smallest non-zero integer such that G_c = 1.
		Each factor group is represented by a special form of polycyclic presentation, a nilpotent presentation, that makes use of the nilpotent structure of the factor group.
		Generators for the c-th lower central factor are defined as commutators of the form [y,x], where x is a generator of weight 1 and y is a generator of weight c-1.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/nq/README.nq (2068 words)

	Nilpotent Ideals (Site not responding. Last check: 2007-11-02)
		Let h be a nilpotent ideal, or nilpotent left ideal if you prefer.
		Since xy lies in h, it is nilpotent.
		Remember that a nil ideal consists of nilpotent elements, even though the entire ideal may not be nilpotent.
	www.mathreference.com /ring-jr,nilj.html (123 words)

	Atlas: Minimal nilpotent algebras in Goursat flags of lengths not exceeding 6 by Piotr Mormul
		Goursat distributions are special rank-2 subbundles, D, in the tangent bundle to an n-dimensional manifold (n >= 4) such that the sequence of consecutive Lie squares of D consists of regular distributions of ranks 3, 4,..., n-1, n.
		Nilpotent local bases, existing only for certain distributions, were addressed systematically in [HLS] and in several posterior works.
		In [M3] computed were the nilpotency orders for the [now known to be nilpotent] bases of G. germs put forward in [KR].
	atlas-conferences.com /cgi-bin/abstract/cajc-15 (984 words)

	AMCA: Nilpotent Lie superalgebras by Marc Gilg
		Nilpotent Lie superalgebras For a Lie superalgebra G over an algebraic closed field of carateristic 0 we define the lower central series C
		The set of filiform Lie superalgebras is an open set of the variety of nilpotent Lie superalgebras' laws.
		Classifications of filiforms over C in lower dimensions Using the bases of the cocycles and adapted chages of bases, like it was done for Lie algebras we have classifications of filiform Lie superalgebras.
	at.yorku.ca /c/a/f/e/09.htm (218 words)

	Amazon.ca: Books: Nilpotent Lie Algebras (Site not responding. Last check: 2007-11-02)
		This volume is devoted to the theory of nilpotent Lie algebras and their applications.
		Nilpotent Lie algebras have played an important role over the last years both in the domain of algebra, considering its role in the classification problems of Lie algebras, and in the domain of differential geometry.
		Audience: This book is intended for graduate students specialising in algebra, differential geometry and in theoretical physics and for researchers in mathematics and in theoretical physics.
	www.amazon.ca /exec/obidos/ASIN/0792339320 (287 words)

	NILPOTENT GROUPS (Site not responding. Last check: 2007-11-02)
		(N1) (A.Miasnikov) Let G be a free nilpotent group of finite rank.
		(N2) Let G be a finitely generated nilpotent group.
		(N4) (G.Baumslag) Let G be a finitely generated torsion-free nilpotent group.
	cs.gc.cuny.edu /~cryptlab/gworld/problems/probnil.html (168 words)

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