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Discussion on Some Properties of an Infinite Non-Abelian Group
International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017 Discussion on Some Then A.B = Properties of an Infinite Non-abelian = Group Where = Ankur Bala#1, Madhuri Kohli#2 #1 M.Sc. , Department of Mathematics, University of Delhi, Delhi #2 M.Sc., Department of Mathematics, University of Delhi, Delhi ( 1 ) India Abstract: In this Article, we have discussed some of the properties of the infinite non-abelian group of matrices whose entries from integers with non-zero determinant. Such as the number of elements of order 2, number of subgroups of order 2 in this group. Moreover for every finite group , there exists such that has a subgroup isomorphic to the group . ( 2 ) Keywords: Infinite non-abelian group, Now from (1) and (2) one can easily say that is Notations: GL(,) n Z []aij n n : aij Z. & homomorphism. det( ) = 1 Now consider , Theorem 1: can be embedded in Clearly Proof: If we prove this theorem for , Hence is injective homomorphism. then we are done. So, by fundamental theorem of isomorphism one can Let us define a mapping easily conclude that can be embedded in for all m . Such that Theorem 2: is non-abelian infinite group Proof: Consider Consider Let A = and And let H= Clearly H is subset of . And H has infinite elements. B= Hence is infinite group. Now consider two matrices Such that determinant of A and B is , where . And ISSN: 2231-5373 http://www.ijmttjournal.org Page 108 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017 Proof: Let G be any group and A(G) be the group of all permutations of set G. -
The Mathematical Structure of the Aesthetic
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 12 June 2017 doi:10.20944/preprints201706.0055.v1 Peer-reviewed version available at Philosophies 2017, 2, 14; doi:10.3390/philosophies2030014 Article A New Kind of Aesthetics – The Mathematical Structure of the Aesthetic Akihiro Kubota 1,*, Hirokazu Hori 2, Makoto Naruse 3 and Fuminori Akiba 4 1 Art and Media Course, Department of Information Design, Tama Art University, 2-1723 Yarimizu, Hachioji, Tokyo 192-0394, Japan; [email protected] 2 Interdisciplinary Graduate School, University of Yamanashi, 4-3-11 Takeda, Kofu,Yamanashi 400-8511, Japan; [email protected] 3 Network System Research Institute, National Institute of Information and Communications Technology, 4-2-1 Nukui-kita, Koganei, Tokyo 184-8795, Japan; [email protected] 4 Philosophy of Information Group, Department of Systems and Social Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan; [email protected] * Correspondence: [email protected]; Tel.: +81-42-679-5634 Abstract: This paper proposes a new approach to investigation into the aesthetics. Specifically, it argues that it is possible to explain the aesthetic and its underlying dynamic relations with axiomatic structure (the octahedral axiom derived category) based on contemporary mathematics – namely, category theory – and through this argument suggests the possibility for discussion about the mathematical structure of the aesthetic. If there was a way to describe the structure of aesthetics with the language of mathematical structures and mathematical axioms – a language completely devoid of arbitrariness – then we would make possible a synthetical argument about the essential human activity of “the aesthetics”, and we would also gain a new method and viewpoint on the philosophy and meaning of the act of creating a work of art and artistic activities. -
Complete Objects in Categories
Complete objects in categories James Richard Andrew Gray February 22, 2021 Abstract We introduce the notions of proto-complete, complete, complete˚ and strong-complete objects in pointed categories. We show under mild condi- tions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This to- gether with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explana- tion behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete. 1 Introduction Recall that Carmichael [19] called a group G complete if it has trivial cen- ter and each automorphism is inner. For each group G there is a canonical homomorphism cG from G to AutpGq, the automorphism group of G. This ho- momorphism assigns to each g in G the inner automorphism which sends each x in G to gxg´1. It can be readily seen that a group G is complete if and only if cG is an isomorphism. Baer [1] showed that a group G is complete if and only if every normal monomorphism with domain G is a split monomorphism. We call an object in a pointed category complete if it satisfies this latter condi- arXiv:2102.09834v1 [math.CT] 19 Feb 2021 tion. -
The Unitary Representations of the Poincaré Group in Any Spacetime
The unitary representations of the Poincar´e group in any spacetime dimension Xavier Bekaert a and Nicolas Boulanger b a Institut Denis Poisson, Unit´emixte de Recherche 7013, Universit´ede Tours, Universit´ed’Orl´eans, CNRS, Parc de Grandmont, 37200 Tours (France) [email protected] b Service de Physique de l’Univers, Champs et Gravitation Universit´ede Mons – UMONS, Place du Parc 20, 7000 Mons (Belgium) [email protected] An extensive group-theoretical treatment of linear relativistic field equa- tions on Minkowski spacetime of arbitrary dimension D > 2 is presented in these lecture notes. To start with, the one-to-one correspondence be- tween linear relativistic field equations and unitary representations of the isometry group is reviewed. In turn, the method of induced representa- tions reduces the problem of classifying the representations of the Poincar´e group ISO(D 1, 1) to the classification of the representations of the sta- − bility subgroups only. Therefore, an exhaustive treatment of the two most important classes of unitary irreducible representations, corresponding to massive and massless particles (the latter class decomposing in turn into the “helicity” and the “infinite-spin” representations) may be performed via the well-known representation theory of the orthogonal groups O(n) (with D 4 <n<D ). Finally, covariant field equations are given for each − unitary irreducible representation of the Poincar´egroup with non-negative arXiv:hep-th/0611263v2 13 Jun 2021 mass-squared. Tachyonic representations are also examined. All these steps are covered in many details and with examples. The present notes also include a self-contained review of the representation theory of the general linear and (in)homogeneous orthogonal groups in terms of Young diagrams. -
Structure” of Physics: a Case Study∗ (Journal of Philosophy 106 (2009): 57–88)
The “Structure” of Physics: A Case Study∗ (Journal of Philosophy 106 (2009): 57–88) Jill North We are used to talking about the “structure” posited by a given theory of physics. We say that relativity is a theory about spacetime structure. Special relativity posits one spacetime structure; different models of general relativity posit different spacetime structures. We also talk of the “existence” of these structures. Special relativity says that the world’s spacetime structure is Minkowskian: it posits that this spacetime structure exists. Understanding structure in this sense seems important for understand- ing what physics is telling us about the world. But it is not immediately obvious just what this structure is, or what we mean by the existence of one structure, rather than another. The idea of mathematical structure is relatively straightforward. There is geometric structure, topological structure, algebraic structure, and so forth. Mathematical structure tells us how abstract mathematical objects t together to form different types of mathematical spaces. Insofar as we understand mathematical objects, we can understand mathematical structure. Of course, what to say about the nature of mathematical objects is not easy. But there seems to be no further problem for understanding mathematical structure. ∗For comments and discussion, I am extremely grateful to David Albert, Frank Arntzenius, Gordon Belot, Josh Brown, Adam Elga, Branden Fitelson, Peter Forrest, Hans Halvorson, Oliver Davis Johns, James Ladyman, David Malament, Oliver Pooley, Brad Skow, TedSider, Rich Thomason, Jason Turner, Dmitri Tymoczko, the philosophy faculty at Yale, audience members at The University of Michigan in fall 2006, and in 2007 at the Paci c APA, the Joint Session of the Aristotelian Society and Mind Association, and the Bellingham Summer Philosophy Conference. -
Group Theory in Particle Physics
Group Theory in Particle Physics Joshua Albert Phy 205 http://en.wikipedia.org/wiki/Image:E8_graph.svg Where Did it Come From? Group Theory has it©s origins in: ● Algebraic Equations ● Number Theory ● Geometry Some major early contributers were Euler, Gauss, Lagrange, Abel, and Galois. What is a group? ● A group is a collection of objects with an associated operation. ● The group can be finite or infinite (based on the number of elements in the group. ● The following four conditions must be satisfied for the set of objects to be a group... 1: Closure ● The group operation must associate any pair of elements T and T© in group G with another element T©© in G. This operation is the group multiplication operation, and so we write: – T T© = T©© – T, T©, T©© all in G. ● Essentially, the product of any two group elements is another group element. 2: Associativity ● For any T, T©, T©© all in G, we must have: – (T T©) T©© = T (T© T©©) ● Note that this does not imply: – T T© = T© T – That is commutativity, which is not a fundamental group property 3: Existence of Identity ● There must exist an identity element I in a group G such that: – T I = I T = T ● For every T in G. 4: Existence of Inverse ● For every element T in G there must exist an inverse element T -1 such that: – T T -1 = T -1 T = I ● These four properties can be satisfied by many types of objects, so let©s go through some examples... Some Finite Group Examples: ● Parity – Representable by {1, -1}, {+,-}, {even, odd} – Clearly an important group in particle physics ● Rotations of an Equilateral Triangle – Representable as ordering of vertices: {ABC, ACB, BAC, BCA, CAB, CBA} – Can also be broken down into subgroups: proper rotations and improper rotations ● The Identity alone (smallest possible group). -
The Frattini Module and P -Automorphisms of Free Pro-P Groups
Publ. RIMS, Kyoto Univ. (), The Frattini module and p!-automorphisms of free pro-p groups. By Darren Semmen ∗ Abstract If a non-trivial subgroup A of the group of continuous automorphisms of a non- cyclic free pro-p group F has finite order, not divisible by p, then the group of fixed points FixF (A) has infinite rank. The semi-direct product F >!A is the universal p-Frattini cover of a finite group G, and so is the projective limit of a sequence of finite groups starting with G, each a canonical group extension of its predecessor by the Frattini module. Examining appearances of the trivial simple module 1 in the Frattini module’s Jordan-H¨older series arose in investigations ([FK97], [BaFr02] and [Sem02]) of modular towers. The number of these appearances prevents FixF (A) from having finite rank. For any group A of automorphisms of a group Γ, the set of fixed points FixΓ(A) := {g ∈ Γ | α(g) = g, ∀α ∈ A} of Γ under the action of A is a subgroup of Γ. Nielsen [N21] and, for the infinite rank case, Schreier [Schr27] showed that any subgroup of a free discrete group will be free. Tate (cf. [Ser02, I.§4.2, Cor. 3a]) extended this to free pro-p groups. In light of this, it is natural to ask for a free group F , what is the rank of FixF (A)? When F is a free discrete group and A is finite, Dyer and Scott [DS75] demonstrated that FixF (A) is a free factor of F , i.e. -
Matrix Lie Groups
Maths Seminar 2007 MATRIX LIE GROUPS Claudiu C Remsing Dept of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 26 September 2007 RhodesUniv CCR 0 Maths Seminar 2007 TALK OUTLINE 1. What is a matrix Lie group ? 2. Matrices revisited. 3. Examples of matrix Lie groups. 4. Matrix Lie algebras. 5. A glimpse at elementary Lie theory. 6. Life beyond elementary Lie theory. RhodesUniv CCR 1 Maths Seminar 2007 1. What is a matrix Lie group ? Matrix Lie groups are groups of invertible • matrices that have desirable geometric features. So matrix Lie groups are simultaneously algebraic and geometric objects. Matrix Lie groups naturally arise in • – geometry (classical, algebraic, differential) – complex analyis – differential equations – Fourier analysis – algebra (group theory, ring theory) – number theory – combinatorics. RhodesUniv CCR 2 Maths Seminar 2007 Matrix Lie groups are encountered in many • applications in – physics (geometric mechanics, quantum con- trol) – engineering (motion control, robotics) – computational chemistry (molecular mo- tion) – computer science (computer animation, computer vision, quantum computation). “It turns out that matrix [Lie] groups • pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry”. (K. Tapp, 2005) RhodesUniv CCR 3 Maths Seminar 2007 EXAMPLE 1 : The Euclidean group E (2). • E (2) = F : R2 R2 F is an isometry . → | n o The vector space R2 is equipped with the standard Euclidean structure (the “dot product”) x y = x y + x y (x, y R2), • 1 1 2 2 ∈ hence with the Euclidean distance d (x, y) = (y x) (y x) (x, y R2). -
Council for Innovative Research Peer Review Research Publishing System
ISSN 2347-3487 Einstein's gravitation is Einstein-Grossmann's equations Alfonso Leon Guillen Gomez Independent scientific researcher, Bogota, Colombia E-mail: [email protected] Abstract While the philosophers of science discuss the General Relativity, the mathematical physicists do not question it. Therefore, there is a conflict. From the theoretical point view “the question of precisely what Einstein discovered remains unanswered, for we have no consensus over the exact nature of the theory's foundations. Is this the theory that extends the relativity of motion from inertial motion to accelerated motion, as Einstein contended? Or is it just a theory that treats gravitation geometrically in the spacetime setting?”. “The voices of dissent proclaim that Einstein was mistaken over the fundamental ideas of his own theory and that their basic principles are simply incompatible with this theory. Many newer texts make no mention of the principles Einstein listed as fundamental to his theory; they appear as neither axiom nor theorem. At best, they are recalled as ideas of purely historical importance in the theory's formation. The very name General Relativity is now routinely condemned as a misnomer and its use often zealously avoided in favour of, say, Einstein's theory of gravitation What has complicated an easy resolution of the debate are the alterations of Einstein's own position on the foundations of his theory”, (Norton, 1993) [1]. Of other hand from the mathematical point view the “General Relativity had been formulated as a messy set of partial differential equations in a single coordinate system. People were so pleased when they found a solution that they didn't care that it probably had no physical significance” (Hawking and Penrose, 1996) [2]. -
LECTURE 12: LIE GROUPS and THEIR LIE ALGEBRAS 1. Lie
LECTURE 12: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups Definition 1.1. A Lie group G is a smooth manifold equipped with a group structure so that the group multiplication µ : G × G ! G; (g1; g2) 7! g1 · g2 is a smooth map. Example. Here are some basic examples: • Rn, considered as a group under addition. • R∗ = R − f0g, considered as a group under multiplication. • S1, Considered as a group under multiplication. • Linear Lie groups GL(n; R), SL(n; R), O(n) etc. • If M and N are Lie groups, so is their product M × N. Remarks. (1) (Hilbert's 5th problem, [Gleason and Montgomery-Zippin, 1950's]) Any topological group whose underlying space is a topological manifold is a Lie group. (2) Not every smooth manifold admits a Lie group structure. For example, the only spheres that admit a Lie group structure are S0, S1 and S3; among all the compact 2 dimensional surfaces the only one that admits a Lie group structure is T 2 = S1 × S1. (3) Here are two simple topological constraints for a manifold to be a Lie group: • If G is a Lie group, then TG is a trivial bundle. n { Proof: We identify TeG = R . The vector bundle isomorphism is given by φ : G × TeG ! T G; φ(x; ξ) = (x; dLx(ξ)) • If G is a Lie group, then π1(G) is an abelian group. { Proof: Suppose α1, α2 2 π1(G). Define α : [0; 1] × [0; 1] ! G by α(t1; t2) = α1(t1) · α2(t2). Then along the bottom edge followed by the right edge we have the composition α1 ◦ α2, where ◦ is the product of loops in the fundamental group, while along the left edge followed by the top edge we get α2 ◦ α1. -
Relativity Without Tears
Vol. 39 (2008) ACTA PHYSICA POLONICA B No 4 RELATIVITY WITHOUT TEARS Z.K. Silagadze Budker Institute of Nuclear Physics and Novosibirsk State University 630 090, Novosibirsk, Russia (Received December 21, 2007) Special relativity is no longer a new revolutionary theory but a firmly established cornerstone of modern physics. The teaching of special relativ- ity, however, still follows its presentation as it unfolded historically, trying to convince the audience of this teaching that Newtonian physics is natural but incorrect and special relativity is its paradoxical but correct amend- ment. I argue in this article in favor of logical instead of historical trend in teaching of relativity and that special relativity is neither paradoxical nor correct (in the absolute sense of the nineteenth century) but the most nat- ural and expected description of the real space-time around us valid for all practical purposes. This last circumstance constitutes a profound mystery of modern physics better known as the cosmological constant problem. PACS numbers: 03.30.+p Preface “To the few who love me and whom I love — to those who feel rather than to those who think — to the dreamers and those who put faith in dreams as in the only realities — I offer this Book of Truths, not in its character of Truth-Teller, but for the Beauty that abounds in its Truth; constituting it true. To these I present the composition as an Art-Product alone; let us say as a Romance; or, if I be not urging too lofty a claim, as a Poem. What I here propound is true: — therefore it cannot die: — or if by any means it be now trodden down so that it die, it will rise again ‘to the Life Everlasting’. -
Alternative Gravitational Theories in Four Dimensions
Alternative Gravitational Theories in Four Dimensionsa Friedrich W. Hehl Institute for Theoretical Physics, University of Cologne, D-50923 K¨oln, Germany email: [email protected] We argue that from the point of view of gauge theory and of an appropriate interpretation of the interferometer experiments with matter waves in a gravitational field, the Einstein- Cartan theory is the best theory of gravity available. Alternative viable theories are general relativity and a certain teleparallelism model. Objections of Ohanian & Ruffini against the Einstein-Cartan theory are discussed. Subsequently we list the papers which were read at the ‘Alternative 4D Session’ and try to order them, at least partially, in the light of the structures discussed. 1 The best alternative theory? I would call general relativity theory8 GR the best available alternative gravitational 21,14 theory and the next best one its teleparallel equivalent GR||. Because of these two theories, at least, it is good to have this alternative session during the Marcel Grossmann Meeting. Let me try to explain why I grant to GR the distinction of being the best alternative theory. After finally having set up special relativity theory in 1905, Einstein subse- quently addressed the question of how to generalize Newton’s gravitational theory as to make it consistent with special relativity, that is, how to reformulate it in a Poincar´ecovariant way. Newton’s theory was a battle tested theory in the realm of our planetary system and under normal laboratory conditions. It has predictive power as it had been shown by the prediction of the existence of the planet Neptune in the last century.