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Topic: Solvable group

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	Encyclopedia: Solvable group
		In mathematics, the derived group (or commutator subgroup) of a group G is the subgroup G1 generated by all the commutators of elements of G; that is, G1 = <[g,h] : g,h in G>.
		In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i.
		Category: Group theory In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a.
	www.nationmaster.com /encyclopedia/Solvable_group (1685 words)

	Solvable group - Wikipedia, the free encyclopedia
		For finite groups, an equivalent definition is that a solvable group is a group with a composition series whose factors are all cyclic groups of prime order.
		The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic to Z itself, it has no composition series, but the normal series {0,Z}, with its only factor group isomorphic to Z, proves that it is in fact solvable.
		But non-abelian groups may or may not be solvable.
	en.wikipedia.org /wiki/Solvable_group (670 words)

	Topological group (Site not responding. Last check: 2007-11-06)
		In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps.
		An example of a topological group which is not a Lie group is given by the rational numbers Q with the topology inherited from R.
		The theory of group representations is almost identical for finite groups and for compact topological groups.
	hallencyclopedia.com /Topological_group (1045 words)

	Solvable groups (Site not responding. Last check: 2007-11-06)
		The next broad class of groups after abelian and nilpotent are the solvable groups.
		A group is solvable if there is some m such that G^(m)={e}, and the least such m is the length of G.
		However, since not every amalgam of solvable groups is embeddable, and known conditions for embeddability are hard to work with, sundry categories of solvable groups seem like a worthwhile place to study dominions.
	www.math.umt.edu /magidin/research/solvable.html (202 words)

	Solvable group: Definition and Links by Encyclopedian.com - All about Solvable group (Site not responding. Last check: 2007-11-06)
		The concept of solvable groups arose to describe a property shared by the automorphism groups of those polynomials whose roots can be expressed using only radicals (square roots, cube roots, etc. and their sums and products).
		More generally, in keeping with Polya's dictum that "if there's a problem you can't figure out, there's a simpler problem you can't figure out", solvable groups can often be used to reduce a conjecture about a complicated group, into a conjecture about a series of groups with simple structure - cyclic groups of prime order.
		However, the result of the Jordan-Holder Theorem[?] is that any two composition series of a group are equivalent, in the sense that the sequence of factor groups in each series are the same, up to rearrangement of their order in the sequence A
	www.encyclopedian.com /so/Solvable-group.html (832 words)

	Subgroup and Factor Group of a Solvable Group (Site not responding. Last check: 2007-11-06)
		Subgroup and Factor Group of a Solvable Group
		Group Chains, Subgroup and Factor Group of a Solvable Group
		The center is solvable, and the quotient group is a smaller p group, hence solvable.
	www.mathreference.com /grp-chain,sub.html (353 words)

	PlanetMath: solvable group
		See Also: derived subgroup, composition series, Galois criterion for solvability of a polynomial by radicals
		This is version 6 of solvable group, born on 2002-01-05, modified 2004-06-23.
		Object id is 1336, canonical name is Solvable.
	planetmath.org /encyclopedia/Solvable.html (54 words)

	Read about Solvable group at WorldVillage Encyclopedia. Research Solvable group and learn about Solvable group here! (Site not responding. Last check: 2007-11-06)
		For finite groups, an equivalent definition is that a solvable group is a group with a
		In keeping with George Polya's dictum that "if there's a problem you can't figure out, there's a simpler problem you can figure out", solvable groups are often useful for reducing a conjecture about a complicated group, into a conjecture about a series of groups with simple structure - cyclic groups of prime order.
		alternating group of degree 5) it follows that every group with order less than 60 is solvable.
	encyclopedia.worldvillage.com /s/b/Solvable_group (567 words)

	Gross, Fletcher (1964-01-01) The 2-length of a finite solvable group. ...
		Gross, Fletcher (1964-01-01) The 2-length of a finite solvable group.
		In this paper the relationship between the 2-length [...] and the 2-exponent [...] of a finite solvable group G is studied.
		The special case of groups satisfying [...] = 2, i.e., groups whose Sylow 2-groups are of exponent 4, is investigated to determine whether [...] in this case.
	etd.caltech.edu /etd/available/etd-09192002-161906 (176 words)

	Solvable group (Site not responding. Last check: 2007-11-06)
		In the history of mathematics, the origins of group theory lie in the search for a proof of the general insolvability of quintic and higher equations, finally realized by Galoistheory.
		The concept of solvable (or soluble) groups arose to describe aproperty shared by the automorphism groups of those polynomials whose roots can be expressed using only radicals (square roots, cube roots, etc., and their sums and products).
		For finite groups, it is equivalent (and useful) to require that a solvable group have a composition series whose factorsare all cyclic of prime order (as every simple, abelian group must be cyclic of prime order).
	www.therfcc.org /solvable-group-210339.html (776 words)

	Series of groups; solvable groups revisited (Site not responding. Last check: 2007-11-06)
		be a sequence of subgroups of the group
		Since we have already defined a solvable group in Section 8.1, we must show that these two definitions are equivalent.
		is, of course, non-abelian, hence is not a solvable group.
	web.usna.navy.mil /~wdj/tonybook/gpthry/node63.html (689 words)

	Talk:Solvable group - Wikipedia, the free encyclopedia
		Moreover, the opposite sentence "if there's a problem you can't figure out, there's a simpler problem you can figure out" is obviously a reformulation from the works of René Descartes.
		"as every simple, abelian group must be cyclic of prime order" seems to be wrong; actually as every simple, abelian group must be products of cyclic groups (may not be of prime order).
		For an abelian group to be simple it must not have any proper non-trivial subgroups, because all its subgroups are normal.
	en.wikipedia.org /wiki/Talk:Solvable_group (158 words)

	Solvable Polynomials
		Its galois group is a quotient of the solvable galois group.
		The quotient of a solvable group is solvable, hence the galois group associated with the polynomial q(x) is solvable.
		It's galois group is g/h, which is solvable, and has a height of n-1 or less.
	www.mathreference.com /fld-slv,poly.html (958 words)

	Embedding group amalgams
		For the class of abelian groups, or even of abelian groups of a given exponent, the result is easy and was well known.
		For example, it is known that they cannot suffice to establish embeddability of even amalgams of nilpotent groups of class two into a solvable group.
		A typical example is Felix Leinen's theorem on embeddability of an amalgam of solvable groups into solvable groups.
	math.berkeley.edu /~magidin/research/embedding.html (785 words)

	ADFS::HD4.$.Work.courses.98-99.Galois.Notes.N5
		So every finite group can by obtained by extending a finite simple group by a smaller group, and that in turn can be got by extending a finite simple group, and so on, until we reach a simple group.
		it is obtained by extending a solvable group by a solvable group.
		Proposition: Subgroups and quotient groups of solvable groups are solvable.
	www.wra1th.plus.com /Galois/N5.html (463 words)

	SOLVABLE GROUPS (Site not responding. Last check: 2007-11-06)
		(S2) (M.I.Kargapolov) The word problem for groups admitting a single defining relation in the variety of all solvable groups of a given derived length.
		Suppose that all of the 5-generator subgroups of G are embeddable in a finitely presented solvable group of bounded derived length.
		Is G embeddable in a finitely presented solvable group?
	zebra.sci.ccny.cuny.edu /web/nygtc/problems/probsol.html (269 words)

	Amalgams of solvable groups (Site not responding. Last check: 2007-11-06)
		Not every amalgam of solvable groups is embeddable into a solvable group; in fact, not even every amalgam of nil-2 groups is embeddable into a solvable group.
		There is one general necessary and sufficient condition known, but unfortunately it is an external condition, invoking the existence of certain undetermined groups and certain maps from our amalgam to those groups satisfying sundry conditions.
		The theorem also gives a bound for the solvability length of the embedding group in terms of the length of the original groups and the length of some of the groups which occur in the conditions.
	www.math.umt.edu /magidin/research/solvableamalg.html (232 words)

	Ramanujan's Solvable Modular Equations
		Indeed both Hermite and Kronecker showed, in the middle of the last century, that the solution of a general quintic may be effected in terms of the solution of the 5th-order modular equation (5.12) and the roots may thus be given in terms of the theta functions.
		The group is quite easy to compute, it is generated by two permutations.
		For p=5,7, and 11 this gives groups of order 20, 42, and 110, respectively, which are obviously solvable and, in fact, for general primes, the construction always produces a solvable group.
	www.cecm.sfu.ca /organics/papers/borwein/paper/html/node13.html (567 words)

	GAP Manual: 37 Group Libraries
		Two permutations groups of the same degree are considered to be equivalent, if there is a renumbering of points, which maps one group into the other one.
		There are a total of 58760 such groups, 1 of size 2, 2 of size 4, 5 of size 8, 14 of size 16, 51 of size 32, 267 of size 64, 2328 of size 128, and 56092 of size 256.
		There are a total of 594 such groups, 1 of size 3, 2 of size 9, 5 of size 27, 15 of size 81, 67 of size 243, and 504 of size 729.
	www.institut.math.jussieu.fr /~jmichel/htm/CHAP037.htm (11708 words)

	Extended Solvable Group (Site not responding. Last check: 2007-11-06)
		Commutator group is very interesting group, but it also seems to be very technical.
		This set is the generalization of commutator group, however it is not always group.
		In the case of solvable group the r-character above is equal to commutativity.
	www.users.bigpond.com /tiddler2/c10508/algebra2.htm (327 words)

	Solvable Group Isomorphism is (Almost) in NP ∩ coNP (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Solvable Group Isomorphism is (Almost) in NP ∩ coNP (2003)
		Abstract: The Group Isomorphism problem consists in deciding whether two input groups G 1 and G 2 given by their multiplication tables are isomorphic.
		We derandomize this protocol for the case of solvable groups showing the following two results: (a) When the input groups are solvable, we give a uniform NP machine for...
	citeseer.ist.psu.edu /654409.html (442 words)

	GAP Manual: 37 Group Libraries
		is a subgroup of the alternating group of degree
		Any integral matrix group G of dimension n is a subgroup of GL_n(Z) as well as of GL_n(Q) and hence lies in some conjugacy class of integral matrix groups under GL_n(Z) and also in some conjugacy class of rational matrix groups under GL_n(Q).
		If the factor group of S by its translation normal subgroup is solvable then this presentation is chosen such that it is a polycyclic power commutator presentation.
	www.maths.may.ie /staff/jmurray/gap_manual/CHAP037.htm (12322 words)

	Solvable Groups (Site not responding. Last check: 2007-11-06)
		[hep-th/9405044] The Role of Solvable Groups in Quantization of Lie Algebras...
		DC MetaData for: Geometry and Dynamics on the Free Solvable Groups...
		Solvable fundamental groups of algebraic varieties and K"ahler manifolds...
	www.scienceoxygen.com /math/273.html (97 words)

	Solvable Group (Site not responding. Last check: 2007-11-06)
		A solvable group is a group whose composition indices are all
		Group having a ``normal series'' such that each ``normal factor'' is
		are not solvable using finite additions, multiplications, divisions, and root extractions.
	www.math.sdu.edu.cn /mathency/math/s/s494.htm (44 words)

	[IRREDSOL] 5 Primitive solvable groups
		with trivial core; the group acts faithfully and primitively on the cosets of such a maximal subgroup.
		package provides functions for translating between primitive solvable groups and irreducible solvable matrix groups, which are described in Section Translating between irreducible solvable matrix groups and primitive solvable groups.
		Moreover, there are functions for finding primitive solvable groups with given properties, see Section Finding primitive pc groups with given properties and Finding primitive solvable permutation groups with given properties.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/irredsol/htm/CHAP005.htm (704 words)

	Solvable Groups, Double Cosets and Isomorphism Theorems (Site not responding. Last check: 2007-11-06)
		Later, we shall return to such groups, and we shall then give an alternate characterization of them and prove more properties related to solvable groups.
		This name comes from the fact that solvable groups are used in a subject called Galois Theory (also a part of algebra but not treated here) to determine whether or not a polynomial equation is solvable in terms of taking
		The reason for this is that the symmetric group Sn is not a solvable group for
	web.usna.navy.mil /~wdj/tonybook/gpthry/node41.html (308 words)

	Citebase - A finitely-presented solvable group with a small quasi-isometry group
		A finitely-presented solvable group with a small quasi-isometry group
		Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic Lie group.
		In addition, we propose a candidate for a polycyclic group whose quasi-isometry group is a solvable real Lie group, and we introduce a candidate for a quasi-isometrically rigid solvable group that is not finitely presented.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0507191 (536 words)

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