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Topic: Nilpotent group

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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/NilpotentGroup.html (0 words)

	PlanetMath: locally nilpotent group
		A locally nilpotent group is a group in which every finitely generated subgroup is nilpotent.
		All nilpotent groups are locally nilpotent, because subgroups of nilpotent groups are nilpotent.
		This is version 4 of locally nilpotent group, born on 2006-02-14, modified 2007-06-13.
	planetmath.org /encyclopedia/LocallyNilpotentGroup.html (0 words)

	Nilpotent group - ExampleProblems.com
		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.
	www.exampleproblems.com /wiki/index.php/Nilpotent_group (0 words)

	Alec Mihailovs
		First, I construct an isomorphism between the categories of(topological) groups of nilpotency class 2 with 2-divisible center and(topological) Lie rings of nilpotency class 2 with 2-divisible center.
		For a finite group G of nilpotency class 2 of oddorder, I construct a basis in its group algebra C[G],parameterized by elements of g* so that the elements of coadjoint orbits form bases of simple two-side ideals of C[G].
		The properties of that correspondence are similar to the properties of the analogous correspondence given by Kirillov's orbit method for nilpotent connected and simply connected Lie groups.
	www.mihailovs.com /Alec (919 words)

	Summary of my results so far (Site not responding. Last check: )
		However, given a finitely generated nilpotent group G and any subgroup H of G, the dominion of H in G in the category of all nilpotent groups is equal to H.
		The dominion of a subgroup H of a nil-2 group G in the variety of all nil-2 groups is equal to the dominion in the variety generated by G.
		In fact, there is a finitely generated group G, and a subgroup H of G, which is abelian of finite rank, such that the dominion of H in G is abelian of infinite rank.
	www.matem.unam.mx /~magidin/research/results.html (722 words)

	Abstracts of papers and preprints (Site not responding. Last check: )
		In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups.
		In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups.
		For special amalgamation bases we show that there are groups which are special bases in varieties of finite exponent but not in the variety of all nil-2 groups, whereas for weak and strong bases we show this is not the case.
	www.matem.unam.mx /~magidin/applic/abspapers.html (713 words)

	[No title] (Site not responding. Last check: )
		The groups that it holds for are a proper 'subset' >> of solvable groups, however.
		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.
		Subject: nilpotent groups Date: Thu, 28 Jan 1999 13:59:06 -0600 (CST) Newsgroups: [missing] To: rusin@vesuvius.math.niu.edu You are almost right with why nilpotent groups are so called.
	www.math.niu.edu /~rusin/known-math/99/nilpotent_gp (234 words)

	[No title] (Site not responding. Last check: )
		In this talk we ask similar questions for the pro-V topology on a free group where V is a variety of finite groups, that is a class of finite groups closed under quotient, subgroup and (finite) direct product.
		The main tool is a representation of a finitely generated subgroup of a free group by a finite automaton of partial bijections on the "geodesic" cosets of the subgroup.
		While our results say this problem is decidable for finite p-groups and finite nilpotent groups, we mention also that the problem of embedding such an automaton into the Cayley graph of a finite p-group or a finite nilpotent group is algorithmically undecidable.
	www.math.technion.ac.il /~techm/20011227121520011227mar (300 words)

	ABSTRACT ALGEBRA ON LINE: Structure of Groups (part 2)
		A finite group is nilpotent if and only if every maximal subgroup is normal.
		Let G be a multiplicative group with a normal subgroup N, and assume that N is abelian.
		A group of order 4 is either cyclic, or else each nontrivial element has order 2, which characterizes the Klein four-group.
	www.math.niu.edu /~beachy/aaol/structure2.html (755 words)

	Normal Structure and Characteristic Subgroups
		The centre of the group G. For nilpotent groups the centre is computed using the centraliser algorithm [Lo98].
		Otherwise, it is computed as the simultaneous fixed point space of the action of the generators of G on the centre of the Fitting subgroup of G [Eic01].
		The series is returned as a sequence of subgroups of G. Note that since polycyclic groups satisfy the ascending chain condition for subgroups, every polycyclic group G has a finite upper central series.
	magma.maths.usyd.edu.au /magma/htmlhelp/text452.htm (829 words)

	Abstract for Quadratic Presentations Paper (Site not responding. Last check: )
		Rational homotopy theory implies that if a nilpotent group group is Kahler, then it's Malcev Lie algebra must be quadratically presented.
		The usual Heisenberg group is not quadratically presented, hence is not Kahler.
		We show that the Heisenberg groups of ranks 4 and 6 are not Kahler.
	www.math.utah.edu /~carlson/research/eprints/abs/quadpres.html (99 words)

	5. Reduction in Nilpotent Group Rings
		be a nilpotent group given by a convergent PCNI-presentation as described in section 2.
		While in the free Abelian group case symmetrized sets and qc-saturation are successfully used to repair the same deficiency such sets in general will not coincide.
		Again for the case of free Abelian groups this definition corresponds to the definition of critical pairs for Laurent polynomials and the resulting t-polynomials are a specialization of these s-polynomials for the integer group ring
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/09/paper_html/node5.html (0 words)

	The Subgroup Structure (Site not responding. Last check: )
		The group H and the element g must belong to a common group.
		Construct the commutator subgroup of groups H and K, where H and K are subgroups of a common group G. Subgroup Constructions Requiring a Nil-po-tent Covering Group
		The maximal normal subgroup of the nilpotent group G that is contained in the subgroup H of G. Normaliser(G, H) : GrpGPC, GrpGPC -> GrpGPC
	www.math.lsu.edu /magma/text450.htm (296 words)

	[GShW:742] (Site not responding. Last check: )
		In this paper we mainly consider the class LN of all locally nilpotent groups.
		However, our main interest is the construction of torsion-free epi-universal groups in LN_lambda, where G in LN_lambda is said to be epi-universal if any group H in LN_lambda is an epimorphic image of G.
		To prove the torsion-freeness of the constructed locally nilpotent group we adjust the well-known commutator collecting process due to P. Hall to our situation.
	www.math.rutgers.edu /pub/shelah/all/742_abs.html (117 words)

	Dwyer: Pre-nilpotent spaces (Site not responding. Last check: )
		The fundamental group of X must be pre-nilpotent, in the sense that its lower central series stabilizes after a finite number of steps.
		Let G be the quotient of the fundamental group of X by the intersection of its lower central series subgroups, so that G acts on Y.
		Then the action of G on each integral homology group of Y must be pre-nilpotent, in the sense that the filtration of each homology group by powers of the augmentation ideal in the group ring of G must stabilize after a finite number of steps.
	www.nd.edu /~wgd/Html/Prenilpotent.Spaces.html (226 words)

	General Group Properties
		A group G is called elementary abelian if it is an abelian p-group of exponent p for some prime p.
		We define a group G on 5 generators a,..., e of infinite order by fixing the commutators of the generators: (b, a) = e^2, (d, c)=e^3 All other pairs of generators commute.
		Since G is nilpotent, we can compute normalisers and centralisers in G. We define the (dihedral) subgroup D3 of G generated by ac and bd and compute its normaliser in G and its centraliser in the (dihedral) subgroup D2 of G generated by c and d.
	www.math.lsu.edu /magma/text451.htm (391 words)

	NILPOTENT GROUPS (Site not responding. Last check: )
		(N1) (A.Miasnikov) Let G be a free nilpotent group of finite rank.
		(N2) Let G be a finitely generated nilpotent group.
		(We say that a group G has genus 1, if every group with the same set of finite homomorphic images as G, is isomorphic to G).
	zebra.sci.ccny.cuny.edu /web/nygtc/problems/probnil.html (168 words)

	GAP Manual: 26 Special Ag Groups
		For a nilpotent group N, the group lambda_2(N) = lambda_2(P_1) cdots lambda_2(P_l) is the Frattini subgroup of N.
		That means w_1 corresponds to the subgroup in the lower nilpotent series and w_2 to the subgroup in the elementary-abelian series of this factor, and w_3 is the prime dividing the order of g_i.
		Since special ag groups are ag groups all functions for ag groups are applicable to special ag groups.
	parallel.rz.uni-mannheim.de /gap/htm/CHAP026.htm (1749 words)

	Abelian, Nilpotent and Soluble Quotient
		Given a subgroup H of the finitely presented group G, this function computes the elementary divisors of the derived quotient group of H. (The coset table T may be used to define H.) This is done by abelianising the Reidemeister-Schreier presentation for H and then proceeding as above.
		A nilpotent quotient algorithm constructs, from a finite presentation of a group, a polycyclic presentation of a nilpotent quotient of the finitely presented group.
		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)

	IRMA Strasbourg - Publication 1999 (Site not responding. Last check: )
		Passi polynomial maps and functors are applied to various problems in nilpotent group theory.
		As an approximation of group cohomology $H^(G,M)$ with coefficients in a nilpotent* module, a theory of polynomial cohomology $P_nH^*(G,M)$ of degre $n\ge 0$ is introduced, by passing to a functorial quotient of the bar resolution which is of finite type over ${\bf Z}$ in all dimensions if $G$ is of finite type.
		The homology group $H_2(G)$ is computed for all 2-step nilpotent groups; an easily evaluable formula is given for the finite case.
	www-irma.u-strasbg.fr /irma/publications/1999/99022.shtml (399 words)

	[No title] (Site not responding. Last check: )
		Any finitely generated nilpotent group is polycyclic and, therefore, has a subnormal series with cyclic factors.
		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.
		Calculations in the nilpotent group are done using a collector from the left without combinatorial collection.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/nq/README.nq (2068 words)

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