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Type of solvable group in mathematics
a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups
Polycyclic_group
Topics referred to by the same term
including: Polycyclic musks Polycyclic aromatic hydrocarbon Chlorinated polycyclic aromatic hydrocarbon Contorted polycyclic aromatic hydrocarbon Polycyclic group
Polycyclic
Mathematical group that can be generated as the set of powers of a single element
trivial group. Every finitely generated abelian group or nilpotent group is polycyclic. Cycle graph (group) Cyclic module Cyclic sieving Circle group (uncountably
Cyclic_group
Chemical compound having multiple closed rings
In the field of organic chemistry, a polycyclic compound is an organic compound featuring several closed rings of atoms, primarily carbon. These ring substructures
Polycyclic_compound
Group with series of normal subgroups where all factors are cyclic
about supersolvable groups: Supersolvable groups are always polycyclic, and hence solvable. Every finitely generated nilpotent group is supersolvable. Every
Supersolvable_group
Topics referred to by the same term
Iwasawa group or modular group, in the study of subgroup lattices, a group whose subgroup lattice is modular virtually polycyclic group, in infinite group theory
M-group
Hydrocarbon composed of multiple aromatic rings
A polycyclic aromatic hydrocarbon (PAH) is any member of a class of organic compounds that is composed of multiple fused aromatic rings. Most are produced
Polycyclic aromatic hydrocarbon
Polycyclic_aromatic_hydrocarbon
Chemical compound
Chrysene is a polycyclic aromatic hydrocarbon (PAH) with the molecular formula C 18H 12 that consists of four fused benzene rings. It is a natural constituent
Chrysene
Chemical compound
Perylene or perilene is a polycyclic aromatic hydrocarbon with the chemical formula C20H12, occurring as a brown solid. It or its derivatives may be carcinogenic
Perylene
Chemical compound
Benz[e]acephenanthrylene is an organic compound with the chemical formula C20H12. It is a polycyclic aromatic hydrocarbon (PAH) made of four benzene rings around a 5-membered
Benz(e)acephenanthrylene
Chemical modification
Complexes of Chromium with Alkynes. 2. Intramolecular Insertions Leading to Polycyclic Lactams". Journal of the American Chemical Society. 115 (23): 10568–10580
Photolabile_protecting_group
Type of mathematical group
group is also residually finite. More examples of groups that are residually finite are free groups, finitely generated nilpotent groups, polycyclic groups
Residually_finite_group
History of a branch of mathematics
semigroups into groups, studied the isomorphism problem of group rings, established the Malçev correspondence for polycyclic groups, and in the 1960s
History_of_group_theory
Group with subnormal series where all factors are abelian
arrangement of classes of groups: cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group. A group G is called virtually
Solvable_group
polynomial rings, group algebras of polycyclic groups, universal enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups. Suppose
Ore_extension
Problem in finite group theory
{\displaystyle G} . For example, polycyclic groups have solvable word problems since the normal form of an arbitrary word in a polycyclic presentation is readily
Word_problem_for_groups
Chemical compound
Dibenzpyrenequinone is a synthetic vat dye. It is a bright yellow solid. It can be produced by cyclization of 1,5-dibenzoylnaphthalene. Dibenzpyrenequinone
Dibenzpyrenequinone
Chemical compound
Benz[a]anthracene or benzo[a]anthracene is a polycyclic aromatic hydrocarbon with the chemical formula C18H12. It is produced during incomplete combustion
Benz(a)anthracene
Organic molecule (C18H12) made of four fused benzene rings
Benzo[c]phenanthrene is a polycyclic aromatic hydrocarbon with the chemical formula C18H12. It is a white solid that is soluble in nonpolar organic solvents
Benzo(c)phenanthrene
Chemical compound
(IUPAC name, 11H-benzo[a]fluorene) is a polycyclic aromatic hydrocarbon (PAH). It is currently listed as a Group 3 carcinogen by the IARC. Benzo[c]fluorene
Benzo(a)fluorene
Mathematical ring with well-behaved ideals
G} is an extension of a Noetherian solvable group (i.e. a polycyclic group) by a finite group, then R [ G ] {\displaystyle R[G]} is two-sided Noetherian
Noetherian_ring
Second homology group of a group
"Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group", Journal of Algebra, 320 (2): 927–944, doi:10.1016/j.jalgebra.2008
Schur_multiplier
Chemical compound
described by some as a polycyclic musk, it is neither primarily a musk odour ingredient, nor does it belong to the polycyclic musk group as defined by the
Cashmeran
Chemical compound
fused pair of benzene rings, making it a simple and rather symmetrical polycyclic aromatic hydrocarbon (PAH). It is the main ingredient of traditional mothballs
Naphthalene
Chemical compound
an organic compound with the chemical formula C20H12. Classified as a polycyclic aromatic hydrocarbon (PAH), it is a colourless solid that is poorly soluble
Benzo(j)fluoranthene
Chemical compound
Benzo[e]pyrene (BeP) is a polycyclic aromatic hydrocarbon with the chemical formula C20H12. It is listed as a Group 3 carcinogen by the IARC. Benzopyrene
Benzo(e)pyrene
Class of chemical compounds
phenylene group. Two isomeric species of benzopyrene are benzo[a]pyrene and the less common benzo[e]pyrene. They belong to the chemical class of polycyclic aromatic
Benzopyrene
Danish shipping and logistics company
this pollution, by a lot of heavy metals and tar substances, so-called polycyclic hydrocarbons (PAHs), are moved from the fuel into the sea instead, and
Maersk
Chemical compound
(DBF) is a polycyclic aromatic hydrocarbon with chemical formula (C6H4)2C=CH2. It is best known as one product from deprotection of the Fmoc group. It can
9-Methylene-fluorene
Chemical compound
Anthanthrone is a synthetic anthraquinone. Its derivative 4,10-dibromoanthanthrone (Pigment Red 168) is a component of some industrial paints. It is prepared
Anthanthrone
Chemical compound
6-Dibenzanthracene is an organic compound with the chemical formula C22H14. It is a polycyclic aromatic hydrocarbon (PAH) made of five fused benzene rings. It is a fused
Dibenz(a,h)anthracene
Compound containing rings with delocalized pi electrons
Saxitoxin Group. The EFSA Journal (2009) 1019, 1-76. Keith, Lawrence H. (2015-03-15). "The Source of U.S. EPA's Sixteen PAH Priority Pollutants". Polycyclic Aromatic
Aromatic_compound
contains free groups of infinite rank. Noetherian groups need not be finite extensions of polycyclic groups. Infinite subgroup series can also be defined
Subgroup_series
Chemical compound
Anthracene is a solid polycyclic aromatic hydrocarbon (PAH) of formula C14H10, consisting of three fused benzene rings. It is a component of coal tar.
Anthracene
Molecule with a ring of bonded atoms
more atoms. The term polycyclic is used when more than one ring appears in a single molecule. Naphthalene is formally a polycyclic compound, but is more
Cyclic_compound
Mathematical concept
infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G
Virtually
Polycyclic aromatic hydrocarbon
Indeno[1,2,3-cd]pyrene is a polycyclic aromatic hydrocarbon (PAH), one of 16 PAHs generally measured in studies of environmental exposure and air pollution
Indeno(1,2,3-cd)pyrene
Carcinogenic compound found in smoke and soot
Benzo[a]pyrene (BaP or B[a]P) is a polycyclic aromatic hydrocarbon (PAH) and the result of incomplete combustion of organic matter at temperatures between
Benzo(a)pyrene
A polyquinane is any polycyclic compound consisting of fused five-membered hydrocarbon rings. If the compound is unsaturated instead of saturated, it is
Polyquinane
Chemical compound
classified it as possibly carcinogenic to humans, grouped into IARC group 2B. Dibenz[a,j]anthracene is a polycyclic aromatic hydrocarbon with five benzene rings
Dibenz(a,j)anthracene
Undecidability theorem in group theory
word-hyperbolic group. Being a torsion-free group. Being a polycyclic group. Being a group with a solvable word problem. Being a residually finite group. Being
Adian–Rabin_theorem
Chemical compound
Dibenzopyrenes are a group of high molecular weight polycyclic aromatic hydrocarbons with the molecular formula C24H14. There are five isomers of dibenzopyrene
Dibenzopyrenes
groups to develop cryptographic protocols. Later several other non-commutative structures like Thompson groups, polycyclic groups, Grigorchuk groups,
Non-commutative_cryptography
Product of incomplete combustion of hydrocarbons
wood, and petroleum coke classified as cokes or char. It can include polycyclic aromatic hydrocarbons and heavy metals like mercury. Soot causes various
Soot
Chemical compound
an organic compound with the chemical formula C20H12. Classified as a polycyclic aromatic hydrocarbon (PAH), it forms pale yellow needles or crystals,
Benzo(k)fluoranthene
Chemical compound
predominant nitrated polycyclic aromatic hydrocarbon (pyrene) emitted in a diesel engine. 1-Nitropyrene is listed as an IARC Group 2B carcinogen, indicating
1-Nitropyrene
Chemical compound
3-Nitrobenzanthrone (3-nitro-7H-benz[de]anthracen-7-one) is a chemical compound emitted in diesel exhaust; it is a potent carcinogen. It produced the highest
3-Nitrobenzanthrone
Hopfian groups are named after Heinz Hopf. Every finite group, by an elementary counting argument. More generally, every polycyclic-by-finite group. Any
Hopfian_group
Chemical compound
Triphenylene is an organic compound with the formula (C6H4)3. It is a flat polycyclic aromatic hydrocarbon (PAH) that has a highly symmetric and planar structure
Triphenylene
Theorems that help decompose a finite group based on prime factors of its order
H in G is also such that [N : H] is divisible by p. In other words, a polycyclic generating system of a Sylow p-subgroup can be found by starting from
Sylow_theorems
Type of synthetic scents
detergents led to its common use, which spurred further development of other polycyclic musks including Galaxolide. 3a,6,6,9a-Tetramethyldodecahydronaphtho-[2
Synthetic_musk
British mathematician
nilpotent groups". Inventiones Mathematicae. 93 (1): 185–223. doi:10.1007/BF01393692. S2CID 122967621. —— (1990). "Decidable properties of polycyclic groups".
Dan_Segal
Chlorinated polycyclic aromatic hydrocarbons (Cl-PAHs) are a group of compounds comprising polycyclic aromatic hydrocarbons with two or more aromatic
Chlorinated polycyclic aromatic hydrocarbon
Chlorinated_polycyclic_aromatic_hydrocarbon
Hydrocarbon compounds composed of rings fused such that the molecule is nonplanar
contorted aromatics, or more precisely contorted polycyclic aromatic hydrocarbons, are polycyclic aromatic hydrocarbons (PAHs) in which the fused aromatic
Contorted_aromatics
Chemical compound
where it was discovered and named by Marcellin Berthelot in 1867. As a polycyclic aromatic hydrocarbon, it is insoluble in water and soluble in many organic
Fluorene
Species of bacterium
the P. aeruginosa group. O'Mahony, MM; Dobson, AD; Barnes, JD; Singleton, I (2006). "The use of ozone in the remediation of polycyclic aromatic hydrocarbon
Pseudomonas_alcaligenes
American mathematician and chess player
Techniques in Polycyclic Groups, Doctor of Philosophy thesis, Rice University, hdl:1911/14599. Passman, Donald S. (2011), The Algebraic Structure of Group Rings
Edward_W._Formanek
Chemical compound
Hexanitrohexaazaisowurtzitane, also called HNIW and CL-20, is a polycyclic nitroamine explosive with the formula C6H6N12O12. It has a better oxidizer-to-fuel
Hexanitrohexaazaisowurtzitane
Class of aromatic substances used in perfumes
major classes: aromatic nitro musks, polycyclic musk compounds, and macrocyclic musk compounds. The first two groups have broad uses in industry ranging
Musk
Chemical compound
organic compound with the formula (C18H8(C6H5)4. It is a red colored polycyclic aromatic hydrocarbon. Because of its distinctive optical and electrical
Rubrene
Class of organic compounds with three rings sharing a single carbon bond
In organic chemistry, propellane is any member of a class of polycyclic hydrocarbons, whose carbon skeleton consists of three rings of carbon atoms sharing
Propellane
Hypothesis about the origin of life
The PAH world hypothesis is a speculative hypothesis that proposes that polycyclic aromatic hydrocarbons (PAHs), known to be abundant in the universe, including
PAH_world_hypothesis
Discrete subgroup in a locally compact topological group
algebraic criterion is that the group be polycyclic. If G {\displaystyle G} is a semisimple linear algebraic group in G L n ( R ) {\displaystyle \mathrm
Lattice_(discrete_subgroup)
Chemical compound
Tetracene, also called naphthacene, is a polycyclic aromatic hydrocarbon. It has the appearance of a pale orange powder. Tetracene is the four-ringed member
Tetracene
first iptycene was reported in 1931 by Erich Clar. Paul Bartlett's research group developed Clar's method and made the first triptycene. Following Bertlett's
Iptycene
Chemical compound
Pyrene is a polycyclic aromatic hydrocarbon (PAH) with the formula C16H10. Consisting of four fused benzene rings, it in a flat aromatic compound. This
Pyrene
Class of chemical compounds
benzofurans are a class of chemical compounds based on the heterocyclic and polycyclic compound benzofuran. Many medicines use the benzofuran core as a scaffold
Substituted_benzofuran
Natural satellites of the planet Saturn
On 6 June 2013, scientists at the IAA-CSIC reported the detection of polycyclic aromatic hydrocarbons in the upper atmosphere of Titan. On 23 June 2014
Moons_of_Saturn
American steel-producing company
as naphthalene. Groundwater at the site was found to be polluted with polycyclic aromatic hydrocarbons and trichloroethylene (TCE). In 2005, the EPA, United
U.S._Steel
Chemical compound
known as acepyrene, is a polycyclic aromatic hydrocarbon (PAH). It has a chemical formula of C18H10. It is a member of the group of pyrenes. It is an aromatic
Cyclopenta(cd)pyrene
Chemical compound
surface-area-to-volume ratio and significantly lower (negligible and non-bioavailable) polycyclic aromatic hydrocarbon (PAH) content. Carbon black is used as a colorant
Carbon_black
Chemical compound
Fluoranthene is a polycyclic aromatic hydrocarbon (PAH). The molecule can be viewed as the fusion of naphthalene and benzene unit connected by a five-membered
Fluoranthene
Chemical compound
of combustion and is a nitrated polycyclic aromatic hydrocarbon (fluorene). 2-Nitrofluorene is listed as an IARC Group 2B carcinogen, indicating it is
2-Nitrofluorene
Polycyclic aromatic hydrocarbon composed of three fused benzene rings
Phenanthrene is a polycyclic aromatic hydrocarbon (PAH) with formula C14H10, consisting of three fused benzene rings. It is a colorless, crystal-like solid
Phenanthrene
hydrocarbons consisting of two cycloalkane rings linked to each other by methylene groups. Each linking face is thus a cyclohexane, which has a boat conformation
Asterane
Genus of Gram-negative bacteria
use as bioremediation agents include: P. alcaligenes, which can degrade polycyclic aromatic hydrocarbons. P. mendocina, which is able to degrade toluene
Pseudomonas
Chemical compound
Coronene (also known as superbenzene and cyclobenzene) is a polycyclic aromatic hydrocarbon (PAH) comprising seven peri-fused benzene rings. Its chemical
Coronene
2010 environmental disaster
Between May and June 2010, the spill waters contained 40 times more polycyclic aromatic hydrocarbons (PAHs) than before the spill. PAHs are often linked
Deepwater_Horizon_oil_spill
Hydrocarbon compound (C6H6)
benzene. In his 1890 paper, Armstrong represented benzene nuclei within polycyclic benzenoids by placing inside the benzene nuclei a letter "C", an abbreviation
Benzene
Traditional Paraguayan and Argentine drink
involves smoking them and for this reason they contain a high number of polycyclic aromatic hydrocarbons (PAHs), such as benzo(a)pyrene, which are carcinogenic
Mate_(drink)
Species of whale
to be directly related to environmental contamination, in this case by polycyclic aromatic hydrocarbons, and coincides with the high incidence of this disease
Beluga_whale
Pain medication of the opiate family
stereochemical complexity and consequent synthetic challenge presented by this polycyclic structure, Michael Freemantle has expressed the view that it is "highly
Morphine
Primary male sex hormone
identified as 17β-hydroxyandrost-4-en-3-one (C19H28O2), a solid polycyclic alcohol with a hydroxyl group at the 17th carbon atom. This also made it obvious that
Testosterone
South American holly plant
carcinogenicity to humans" (group 3). Because the traditional preparation of yerba mate leaves involves smoking them, exposure to polycyclic aromatic hydrocarbons
Yerba_mate
Life arising from non-living matter
year. Currently 40,000 tons of cosmic dust falls to Earth each year. Polycyclic aromatic hydrocarbons (PAH) are the most common and abundant polyatomic
Abiogenesis
Type of water pipe
does relatively little to reduce the volatile organic compounds (VOCs), polycyclic aromatic hydrocarbons (PAHs), and carbonyl compounds that represent the
Hookah
2017 fatal fire in West London
significant concentrations of benzene, benzo(a)pyrene, phosphorus and polycyclic aromatic hydrocarbons. Chemicals in the soil are unlikely to seep into
Grenfell_Tower_fire
{G}}(3,1)} of coclass graphs provide an opportunity to give a parametrized polycyclic power-commutator presentation for the complete coclass tree T 1 ( R )
Descendant tree (group theory)
Descendant_tree_(group_theory)
Visual grouping of five galaxies
formation stars. NGC 7317 has the lowest dust mass and it has lowest polycyclic aromatic hydrocarbon (PAH). There is an infrared bright region within
Stephan's_Quintet
Cyclacenes are hoop-like polycyclic compounds where aromatic moieties are fused together to form the ring structures. Because of their interesting electronic
Cyclacene
Chemical compound
Hexa-peri-hexabenzocoronene (HBC) is a polycyclic aromatic hydrocarbon with the molecular formula C42H18. It consists of a central coronene molecule, with
Hexabenzocoronene
Isoelectronic structure
ligands in many applications, such as polycyclic aromatic hydrocarbons, and more commonly low oxidation state main group element bonding. Ever since the first
N-Heterocyclic carbene boryl anion
N-Heterocyclic_carbene_boryl_anion
Chemical property
Contributing atoms arranged in one or more rings. (Cyclic, Bicyclic, or Polycyclic) A number of π delocalized electrons that is even, but not a multiple
Aromaticity
Chemical compound
Sanguinarine is a polycyclic quaternary alkaloid. It is extracted from some plants, including the bloodroot plant (Sanguinaria canadensis, from whose scientific
Sanguinarine
Chemical compound
Kekulene is a polycyclic aromatic hydrocarbon which consists of 12 fused benzene rings arranged in a circle. It is therefore classified as a [12]-circulene
Kekulene
{\displaystyle 5} -groups. The 2 {\displaystyle 2} -groups of coclass 1 {\displaystyle 1} in Figure 5 can be defined by the following parametrized polycyclic pc-presentation
Artin_transfer_(group_theory)
Central nervous system stimulant
tobacco has been shown to increase caffeine clearance by 56% as a result of polycyclic aromatic hydrocarbons inducing the CYP1A2 enzyme. The CYP1A2 enzyme that
Caffeine
Theorem in group theory
for semi-splittings over virtually polycyclic subgroups. Here the case of semi-splittings of word-hyperbolic groups over two-ended (virtually infinite
Stallings theorem about ends of groups
Stallings_theorem_about_ends_of_groups
Sixth planet from the Sun
On 6 June 2013, scientists at the IAA-CSIC reported the detection of polycyclic aromatic hydrocarbons in the upper atmosphere of Titan, a possible precursor
Saturn
Lung disease involving long-term poor airflow
cannabis and tobacco smoke contain toxic and carcinogenic compounds such as polycyclic aromatic hydrocarbons (PAHs), volatile organic compounds (VOCs), and reactive
Chronic obstructive pulmonary disease
Chronic_obstructive_pulmonary_disease
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POLYCYCLIC GROUP
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POLYCYCLIC GROUP
POLYCYCLIC GROUP
POLYCYCLIC GROUP
POLYCYCLIC GROUP
POLYCYCLIC GROUP
POLYCYCLIC GROUP
POLYCYCLIC GROUP
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