On Generalizations of Sylow Tower Groups
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The General Linear Group
18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices. -
Normal Subgroup Structure of Totally Disconnected Locally Compact Groups
Normal subgroup structure of totally disconnected locally compact groups Colin D. Reid Abstract The present article is a summary of joint work of the author and Phillip Wesolek on the normal subgroup structure of totally disconnected locally compact second-countable (t.d.l.c.s.c.) groups. The general strategy is as follows: We obtain normal series for a t.d.l.c.s.c. group in which each factor is ‘small’ or a non-abelian chief factor; we show that up to a certain equivalence relation (called association), a given non-abelian chief factor can be inserted into any finite normal series; and we obtain restrictions on the structure of chief factors, such that the restrictions are invariant under association. Some limitations of this strategy and ideas for future work are also discussed. 1 Introduction A common theme throughout group theory is the reduction of problems concern- ing a group G to those concerning the normal subgroup N and the quotient G=N, where both N and G=N have some better-understood structure; more generally, one can consider a decomposition of G via normal series. This approach has been espe- cially successful for the following classes of groups: finite groups, profinite groups, algebraic groups, connected Lie groups and connected locally compact groups. To summarise the situation for these classes, let us recall the notion of chief factors and chief series. Definition 1. Let G be a Hausdorff topological group. A chief factor K=L of G is a pair of closed normal subgroups L < K such that there are no closed normal subgroups of G lying strictly between K and L.A descending chief series for G is a Colin D. -
Normal Subgroups of the General Linear Groups Over Von Neumann Regular Rings L
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 96, Number 2, February 1986 NORMAL SUBGROUPS OF THE GENERAL LINEAR GROUPS OVER VON NEUMANN REGULAR RINGS L. N. VASERSTEIN1 ABSTRACT. Let A be a von Neumann regular ring or, more generally, let A be an associative ring with 1 whose reduction modulo its Jacobson radical is von Neumann regular. We obtain a complete description of all subgroups of GLn A, n > 3, which are normalized by elementary matrices. 1. Introduction. For any associative ring A with 1 and any natural number n, let GLn A be the group of invertible n by n matrices over A and EnA the subgroup generated by all elementary matrices x1'3, where 1 < i / j < n and x E A. In this paper we describe all subgroups of GLn A normalized by EnA for any von Neumann regular A, provided n > 3. Our description is standard (see Bass [1] and Vaserstein [14, 16]): a subgroup H of GL„ A is normalized by EnA if and only if H is of level B for an ideal B of A, i.e. E„(A, B) C H C Gn(A, B). Here Gn(A, B) is the inverse image of the center of GL„(,4/S) (when n > 2, this center consists of scalar invertible matrices over the center of the ring A/B) under the canonical homomorphism GL„ A —►GLn(A/B) and En(A, B) is the normal subgroup of EnA generated by all elementary matrices in Gn(A, B) (when n > 3, the group En(A, B) is generated by matrices of the form (—y)J'lx1'Jy:i''1 with x € B,y £ A,l < i ^ j < n, see [14]). -
Variational Problems on Flows of Diffeomorphisms for Image Matching
QUARTERLY OF APPLIED MATHEMATICS VOLUME LVI, NUMBER 3 SEPTEMBER 1998, PAGES 587-600 VARIATIONAL PROBLEMS ON FLOWS OF DIFFEOMORPHISMS FOR IMAGE MATCHING By PAUL DUPUIS (LCDS, Division of Applied Mathematics, Brown University, Providence, RI), ULF GRENANDER (Division of Applied Mathematics, Brown University, Providence, Rl), AND MICHAEL I. MILLER (Dept. of Electrical Engineering, Washington University, St. Louis, MO) Abstract. This paper studies a variational formulation of the image matching prob- lem. We consider a scenario in which a canonical representative image T is to be carried via a smooth change of variable into an image that is intended to provide a good fit to the observed data. The images are all defined on an open bounded set GcR3, The changes of variable are determined as solutions of the nonlinear Eulerian transport equation ==v(rj(s;x),s), r)(t;x)=x, (0.1) with the location 77(0;x) in the canonical image carried to the location x in the deformed image. The variational problem then takes the form arg mm ;||2 + [ |Tor?(0;a;) - D(x)\2dx (0.2) JG where ||v|| is an appropriate norm on the velocity field v(-, •), and the second term at- tempts to enforce fidelity to the data. In this paper we derive conditions under which the variational problem described above is well posed. The key issue is the choice of the norm. Conditions are formulated under which the regularity of v(-, ■) imposed by finiteness of the norm guarantees that the associated flow is supported on a space of diffeomorphisms. The problem (0.2) can Received March 15, 1996. -
Let G Be a Group. a Subnormal Series of Subgroups of G Is a Sequence G {E} < G 1 < ... < Gn = G Such That G I−1
NORMAL AND SUBNORMAL SERIES Let G be a group. A subnormal series of subgroups of G is a sequence G0 = feg < G1 < ::: < Gn = G such that Gi−1 / Gi for i = 1; :::; n. A normal series is a subnormal series such that each Gi is normal in G. The number n is called the length of the (sub)normal series. A (sub)normal series is called proper if all the quotient groups Gi=Gi−1 are nontrivial. These quotient groups are refered to as successive quotients of the (sub)normal series. We say that two (sub)normal series are equivalent if they have the same length and there is a permutation π 2 Sn such that the groups Gi=Gi−1 and Gπ(i)=Gπ(i)−1 are isomorphic for all i = 1; 2; :::; n. A refinement of a (sub)normal series G0 = feg < G1 < ::: < Gn = G is a 0 0 0 (sub)normal series G0 = feg < G1 < ::: < Gm = G such that there is a strictly 0 increasing function f : f1; :::; ng −! f1; 2; :::; mg such that Gi = Gf(i) for all i. We prove now the following important theorem: Schreier Refinement Theorem. Any two (sub)normal series in G have equivalent refinements. It the series are proper, the refinements can be chosen proper as well. Proof: Note first that each integer k can be uniquely written in the form (m+1)i+j for some integer i and some j 2 f0; :::; mg. If 0 ≤ k < (m+1)(n+1) then 0 ≤ i ≤ n. Similarly, each integer 0 ≤ k < (m + 1)(n + 1) can be uniquely written in the form (n + 1)j + i, where 0 ≤ i ≤ n and 0 ≤ j ≤ m. -
LM2931-N SNOSBE5G –MARCH 2000–REVISED APRIL 2013 LM2931-N Series Low Dropout Regulators Check for Samples: LM2931-N
LM2931-N www.ti.com SNOSBE5G –MARCH 2000–REVISED APRIL 2013 LM2931-N Series Low Dropout Regulators Check for Samples: LM2931-N 1FEATURES DESCRIPTION The LM2931-N positive voltage regulator features a 2• Very Low Quiescent Current very low quiescent current of 1mA or less when • Output Current in Excess of 100 mA supplying 10mA loads. This unique characteristic and • Input-output Differential Less than 0.6V the extremely low input-output differential required for • Reverse Battery Protection proper regulation (0.2V for output currents of 10mA) make the LM2931-N the ideal regulator for standby • 60V Load Dump Protection power systems. Applications include memory standby • −50V Reverse Transient Protection circuits, CMOS and other low power processor power • Short Circuit Protection supplies as well as systems demanding as much as • Internal Thermal Overload Protection 100mA of output current. • Mirror-image Insertion Protection Designed originally for automotive applications, the LM2931-N and all regulated circuitry are protected • Available in TO-220, TO-92, TO-263, or SOIC-8 from reverse battery installations or 2 battery jumps. Packages During line transients, such as a load dump (60V) • Available as Adjustable with TTL Compatible when the input voltage to the regulator can Switch momentarily exceed the specified maximum operating voltage, the regulator will automatically shut down to protect both internal circuits and the load. The LM2931-N cannot be harmed by temporary mirror-image insertion. Familiar regulator features such as short circuit and thermal overload protection are also provided. The LM2931-N family includes a fixed 5V output (±3.8% tolerance for A grade) or an adjustable output with ON/OFF pin. -
Extreme Classes of Finite Soluble Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 9, 285-313 (1968) Extreme Classes of Finite Soluble Groups ROGER CARTER The Mathematics Institute, University of Warwick, Coventry, England BERND FISCHER Mathematisches Seminar der Universitdt Frankfurt am Main, Germany TREVOR HAWKES Corpus Christi College, Cambridge, England Communicated by B. Huppert Received August 24, 1967 Knowledge of the structure of a specified collection 9 of subgroups of a group G can often yield detailed information about G itself. For example we have the elementary fact that if all the minimal cyclic subgroups of a group G are n-groups then G is a n-group. Another example is provided by an unpublished theorem of M. B. Powell who has shown that if we take for 9’ the collection of 2-generator subgroups of a finite soluble group G then G has p-length at most n if and only if every member of Y has p-length at most n. Clearly one should aim to make the set Y as small as possible con- sistent with still retrieving useful information about the structure of G. In this note we investigate a situation of this type and for the purpose intro- duce a class (3 of extreme groups. (5 is a subclass of the class of finite soluble groups and loosely speaking comprises those groups which contain as few complemented chief factors, as possible, in a given chief series; more precisely, a group G is extreme if it has a chief series with exactly Z(G) complemented chief factors, where Z(G) denotes the nilpotent (or Fitting) length of G. -
Quasi P Or Not Quasi P? That Is the Question
Rose-Hulman Undergraduate Mathematics Journal Volume 3 Issue 2 Article 2 Quasi p or not Quasi p? That is the Question Ben Harwood Northern Kentucky University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Harwood, Ben (2002) "Quasi p or not Quasi p? That is the Question," Rose-Hulman Undergraduate Mathematics Journal: Vol. 3 : Iss. 2 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol3/iss2/2 Quasi p- or not quasi p-? That is the Question.* By Ben Harwood Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 e-mail: [email protected] Section Zero: Introduction The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prove that (Z3 £Z3)oZ2 and Z5 o Z4 are quasi 2-groups. Those proofs now form a portion of Proposition (12.1) It should also be noted that [Hwd] may also be found in this journal. Section One: Why should we be interested in quasi p-groups? In a 1957 paper titled Coverings of algebraic curves [Abh2], Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field k of prime characteristic p is the set of quasi p-groups, where by the algebraic fundamental group of the affine line he meant the family of all Galois groups Gal(L=k(X)) as L varies over all finite normal extensions of k(X) the function field of the affine line such that no point of the line is ramified in L, and where by a quasi p-group he meant a finite group that is generated by all of its p-Sylow subgroups. -
A Fast Diffeomorphic Image Registration Algorithm
www.elsevier.com/locate/ynimg NeuroImage 38 (2007) 95–113 A fast diffeomorphic image registration algorithm John Ashburner Wellcome Trust Centre for Neuroimaging, 12 Queen Square, London, WC1N 3BG, UK Received 26 October 2006; revised 14 May 2007; accepted 3 July 2007 Available online 18 July 2007 This paper describes DARTEL, which is an algorithm for diffeo- Many registration approaches still use a small deformation morphic image registration. It is implemented for both 2D and 3D model. These models parameterise a displacement field (u), which image registration and has been formulated to include an option for is simply added to an identity transform (x). estimating inverse consistent deformations. Nonlinear registration is considered as a local optimisation problem, which is solved using a Φ x x u x 1 Levenberg–Marquardt strategy. The necessary matrix solutions are ð Þ ¼ þ ð Þ ð Þ obtained in reasonable time using a multigrid method. A constant In such parameterisations, the inverse transformation is sometimes Eulerian velocity framework is used, which allows a rapid scaling and approximated by subtracting the displacement. It is worth noting that squaring method to be used in the computations. DARTEL has been this is only a very approximate inverse, which fails badly for larger applied to intersubject registration of 471 whole brain images, and the deformations. As shown in Fig. 1, compositions of these forward resulting deformations were evaluated in terms of how well they encode and “inverse” deformations do not produce an identity transform. the shape information necessary to separate male and female subjects Small deformation models do not necessarily enforce a one-to-one and to predict the ages of the subjects. -
Deep Compositing Using Lie Algebras
Deep Compositing Using Lie Algebras TOM DUFF Pixar Animation Studios Deep compositing is an important practical tool in creating digital imagery, Deep images extend Porter-Duff compositing by storing multiple but there has been little theoretical analysis of the underlying mathematical values at each pixel with depth information to determine composit- operators. Motivated by finding a simple formulation of the merging oper- ing order. The OpenEXR deep image representation stores, for each ation on OpenEXR-style deep images, we show that the Porter-Duff over pixel, a list of nonoverlapping depth intervals, each containing a function is the operator of a Lie group. In its corresponding Lie algebra, the single pixel value [Kainz 2013]. An interval in a deep pixel can splitting and mixing functions that OpenEXR deep merging requires have a represent contributions either by hard surfaces if the interval has particularly simple form. Working in the Lie algebra, we present a novel, zero extent or by volumetric objects when the interval has nonzero simple proof of the uniqueness of the mixing function. extent. Displaying a single deep image requires compositing all the The Lie group structure has many more applications, including new, values in each deep pixel in depth order. When two deep images correct resampling algorithms for volumetric images with alpha channels, are merged, splitting and mixing of these intervals are necessary, as and a deep image compression technique that outperforms that of OpenEXR. described in the following paragraphs. 26 r Merging the pixels of two deep images requires that correspond- CCS Concepts: Computing methodologies → Antialiasing; Visibility; ing pixels of the two images be reconciled so that all overlapping Image processing; intervals are identical in extent. -
Integer Discrete Flows and Lossless Compression
Integer Discrete Flows and Lossless Compression Emiel Hoogeboom⇤ Jorn W.T. Peters⇤ UvA-Bosch Delta Lab UvA-Bosch Delta Lab University of Amsterdam University of Amsterdam Netherlands Netherlands [email protected] [email protected] Rianne van den Berg† Max Welling University of Amsterdam UvA-Bosch Delta Lab Netherlands University of Amsterdam [email protected] Netherlands [email protected] Abstract Lossless compression methods shorten the expected representation size of data without loss of information, using a statistical model. Flow-based models are attractive in this setting because they admit exact likelihood optimization, which is equivalent to minimizing the expected number of bits per message. However, conventional flows assume continuous data, which may lead to reconstruction errors when quantized for compression. For that reason, we introduce a flow-based generative model for ordinal discrete data called Integer Discrete Flow (IDF): a bijective integer map that can learn rich transformations on high-dimensional data. As building blocks for IDFs, we introduce a flexible transformation layer called integer discrete coupling. Our experiments show that IDFs are competitive with other flow-based generative models. Furthermore, we demonstrate that IDF based compression achieves state-of-the-art lossless compression rates on CIFAR10, ImageNet32, and ImageNet64. To the best of our knowledge, this is the first lossless compression method that uses invertible neural networks. 1 Introduction Every day, 2500 petabytes of data are generated. Clearly, there is a need for compression to enable efficient transmission and storage of this data. Compression algorithms aim to decrease the size of representations by exploiting patterns and structure in data. -
Computing the Scale of an Endomorphism of a Totally Disconnected Locally Compact Group
axioms Article Computing the Scale of an Endomorphism of a totally Disconnected Locally Compact Group George A. Willis School of Mathematical and Physical Sciences, University of Newcastle, University Drive, Callaghan NSW 2308, Australia; [email protected]; Tel.: +61-2-4921-5666; Fax: +61-2-4921-6898 Received: 28 August 2017; Accepted: 9 October 2017; Published: 20 October 2017 Abstract: The scale of an endomorphism of a totally disconnected, locally compact group G is defined and an example is presented which shows that the scale function is not always continuous with respect to the Braconnier topology on the automorphism group of G. Methods for computing the scale, which is a positive integer, are surveyed and illustrated by applying them in diverse cases, including when G is compact; an automorphism group of a tree; Neretin’s group of almost automorphisms of a tree; and a p-adic Lie group. The information required to compute the scale is reviewed from the perspective of the, as yet incomplete, general theory of totally disconnected, locally compact groups. Keywords: locally compact group; endomorphism; scale; tree; Neretin’s group; Thompson’s group; p-adic Lie group 1. Introduction Let G be a totally disconnected, locally compact (t.d.l.c.) group. A fundamental theorem about t.d.l.c. groups, proved by van Dantzig in the 1930s, see [1] and ([2] Theorem II.7.7), asserts that G has a base of neighbourhoods of the identity consisting of compact open subgroups. These subgroups are important for the definition of the scale of endomorphisms a : G ! G, which is a positive integer gauging the action of a.