61 research outputs found
Length-type parameters of finite groups with almost unipotent automorphisms
Let be an automorphism of a finite group . For a positive integer , let be the subgroup generated by all commutators in the semidirect product over , where is repeated times. By Baer's theorem, if , then the commutator subgroup is nilpotent. We generalize this theorem in terms of certain length parameters of . For soluble we prove that if, for some , the Fitting height of is equal to , then the Fitting height of is at most . For nonsoluble the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height of a finite group is the least number such that , where , and is the inverse image of the generalized Fitting subgroup . Let be the number of prime factors of the order counting multiplicities. It is proved that if, for some , the generalized Fitting height of is equal to , then the generalized Fitting height of is bounded in terms of and .
The nonsoluble length~ of a finite group~ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if , then the nonsoluble length of is bounded in terms of and .
We also state conjectures of stronger results independent of and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups
Nonsoluble and non-p-soluble length of finite groups
Every finite group G has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length λ(G) as the number of nonsoluble factors in a shortest series of this kind. Upper bounds for λ(G) appear in the study of various problems on finite, residually finite, and profinite groups. We prove that λ(G) is bounded in terms of the maximum 2-length of soluble subgroups of G, and that λ(G) is bounded by the maximum Fitting height of soluble subgroups. For an odd prime p, the non-p-soluble length λ p (G) is introduced, and it is proved that λ p (G) does not exceed the maximum p-length of p-soluble subgroups. We conjecture that for a given prime p and a given proper group variety V the non-p-soluble length λ p (G) of finite groups G whose Sylow p-subgroups belong to V is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent. As an application of the results obtained, an error is corrected in the proof of the main result of the second authorâs paper Multilinear commutators in residually finite groups, Israel Journal of Mathematics 189 (2012), 207â224
On the algebraic structures connected with the linear Poisson brackets of hydrodynamics type
The generalized form of the Kac formula for Verma modules associated with
linear brackets of hydrodynamics type is proposed. Second cohomology groups of
the generalized Virasoro algebras are calculated. Connection of the central
extensions with the problem of quntization of hydrodynamics brackets is
demonstrated
Classical Monopoles: Newton, NUT-space, gravomagnetic lensing and atomic spectra
Stimulated by a scholium in Newton's Principia we find some beautiful results
in classical mechanics which can be interpreted in terms of the orbits in the
field of a mass endowed with a gravomagnetic monopole. All the orbits lie on
cones! When the cones are slit open and flattened the orbits are exactly the
ellipses and hyperbolae that one would have obtained without the gravomagnetic
monopole.
The beauty and simplicity of these results has led us to explore the similar
problems in Atomic Physics when the nuclei have an added Dirac magnetic
monopole. These problems have been explored by others and we sketch the
derivations and give details of the predicted spectrum of monopolar hydrogen.
Finally we return to gravomagnetic monopoles in general relativity. We
explain why NUT space has a non-spherical metric although NUT space itself is
the spherical space-time of a mass with a gravomagnetic monopole. We
demonstrate that all geodesics in NUT space lie on cones and use this result to
study the gravitational lensing by bodies with gravomagnetic monopoles.
We remark that just as electromagnetism would have to be extended beyond
Maxwell's equations to allow for magnetic monopoles and their currents so
general relativity would have to be extended to allow torsion for general
distributions of gravomagnetic monopoles and their currents. Of course if
monopoles were never discovered then it would be a triumph for both Maxwellian
Electromagnetism and General Relativity as they stand!Comment: 39 pages, 9 figures and 2 tables available on request from the
author
Physically motivated uncertainty relations at the Planck length for an emergent non commutative spacetime
We derive new space-time uncertainty relations (STUR) at the fundamental
Planck length from quantum mechanics and general relativity (GR), both in
flat and curved backgrounds. Contrary to claims present in the literature, our
approach suggests that no minimal uncertainty appears for lengths, but instead
for minimal space and four-volumes. Moreover, we derive a maximal absolute
value for the energy density. Finally, some considerations on possible
commutators among quantum operators implying our STUR are done.Comment: Final version published in "Class. Quantum Grav.
Division Algebras and Quantum Theory
Quantum theory may be formulated using Hilbert spaces over any of the three
associative normed division algebras: the real numbers, the complex numbers and
the quaternions. Indeed, these three choices appear naturally in a number of
axiomatic approaches. However, there are internal problems with real or
quaternionic quantum theory. Here we argue that these problems can be resolved
if we treat real, complex and quaternionic quantum theory as part of a unified
structure. Dyson called this structure the "three-fold way". It is perhaps
easiest to see it in the study of irreducible unitary representations of groups
on complex Hilbert spaces. These representations come in three kinds: those
that are not isomorphic to their own dual (the truly "complex"
representations), those that are self-dual thanks to a symmetric bilinear
pairing (which are "real", in that they are the complexifications of
representations on real Hilbert spaces), and those that are self-dual thanks to
an antisymmetric bilinear pairing (which are "quaternionic", in that they are
the underlying complex representations of representations on quaternionic
Hilbert spaces). This three-fold classification sheds light on the physics of
time reversal symmetry, and it already plays an important role in particle
physics. More generally, Hilbert spaces of any one of the three kinds - real,
complex and quaternionic - can be seen as Hilbert spaces of the other kinds,
equipped with extra structure.Comment: 30 pages, 3 encapsulated Postscript figure
Growth, entropy and commutativity of algebras satisfying prescribed relations
In 1964, Golod and Shafarevich found that, provided that the number of
relations of each degree satisfy some bounds, there exist infinitely
dimensional algebras satisfying the relations. These algebras are called
Golod-Shafarevich algebras. This paper provides bounds for the growth function
on images of Golod-Shafarevich algebras based upon the number of defining
relations. This extends results from [32], [33]. Lower bounds of growth for
constructed algebras are also obtained, permitting the construction of algebras
with various growth functions of various entropies. In particular, the paper
answers a question by Drensky [7] by constructing algebras with subexponential
growth satisfying given relations, under mild assumption on the number of
generating relations of each degree. Examples of nil algebras with neither
polynomial nor exponential growth over uncountable fields are also constructed,
answering a question by Zelmanov [40].
Recently, several open questions concerning the commutativity of algebras
satisfying a prescribed number of defining relations have arisen from the study
of noncommutative singularities. Additionally, this paper solves one such
question, posed by Donovan and Wemyss in [8].Comment: arXiv admin note: text overlap with arXiv:1207.650
Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics
We examine some of Connes' criticisms of Robinson's infinitesimals starting
in 1995. Connes sought to exploit the Solovay model S as ammunition against
non-standard analysis, but the model tends to boomerang, undercutting Connes'
own earlier work in functional analysis. Connes described the hyperreals as
both a "virtual theory" and a "chimera", yet acknowledged that his argument
relies on the transfer principle. We analyze Connes' "dart-throwing" thought
experiment, but reach an opposite conclusion. In S, all definable sets of reals
are Lebesgue measurable, suggesting that Connes views a theory as being
"virtual" if it is not definable in a suitable model of ZFC. If so, Connes'
claim that a theory of the hyperreals is "virtual" is refuted by the existence
of a definable model of the hyperreal field due to Kanovei and Shelah. Free
ultrafilters aren't definable, yet Connes exploited such ultrafilters both in
his own earlier work on the classification of factors in the 1970s and 80s, and
in his Noncommutative Geometry, raising the question whether the latter may not
be vulnerable to Connes' criticism of virtuality. We analyze the philosophical
underpinnings of Connes' argument based on Goedel's incompleteness theorem, and
detect an apparent circularity in Connes' logic. We document the reliance on
non-constructive foundational material, and specifically on the Dixmier trace
(featured on the front cover of Connes' magnum opus) and the Hahn-Banach
theorem, in Connes' own framework. We also note an inaccuracy in Machover's
critique of infinitesimal-based pedagogy.Comment: 52 pages, 1 figur
- âŠ