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A Study of Some Properties of Generalized Groups OCTOGON MATHEMATICAL MAGAZINE Vol. 17, No.2, October 2009, pp 599-626 599 ISSN 1222-5657, ISBN 978-973-88255-5-0, www.hetfalu.ro/octogon A study of some properties of generalized groups Akinmoyewa, Joseph Toyin9 1. INTRODUCTION Generalized group is an algebraic structure which has a deep physical background in the unified gauge theory and has direct relation with isotopies. 1.1. THE IDEA OF UNIFIED THEORY The unified theory has a direct relation with the geometry of space. It describes particles and their interactions in a quantum mechanical manner and the geometry of the space time through which they are moving. For instance, the result of experiment at very high energies in accelerators such as the Large Electron Position Collider [LEP] at CERN (the European Laboratory for Nuclear Research) in general has led to the discovery of a wealth of so called ’elementary’ particles and that this variety of particles interact with each other via the weak electromagnetic and strong forces(the weak and electromagnetic having been unified). A simple application of Heisenberg’s on certainty principle-the embodiment of quantum mechanics-at credibly small distances leads to the conclusion that at distance scale of around 10−35 metres(the plank length) there are huge fluctuation of energy that sufficiently big to create tiny black holes. Space is not empty but full of these tiny black holes fluctuations that come and go on time scale of around 10−35 seconds(the plank time). The vacuum is full of space-time foam. Currently, the most promising is super-string theory in which the so called ’elementary’ particles are described as vibration of tiny(planck-length)closed loops of strings. In this theory the classical law of physics, such as electromagnetism and general relativity, are modified at time distances comparable to the length of the string. This notion of ’quantum space-time’ is the goal of unified theory of physical forces. 9Received: 21.04.2009 2000 Mathematics Subject Classification. 05E18. Key words and phrases. Groups etc. 600 Akinmoyewa, Joseph Toyin The unified theory offers a new insight into the structure, order and measures of the quantum world of the entire universe. It is in accordance with the quantum mechanics, quantum theory of space time and quantum gravity. It is in conformity with the proposition of Einsteins theory of relativity that the laws of nature and the geometry of space-time are independent. It was proposed that: (i) There is conceptual unification and reconciliation of the theory of relativity with quantum theory and the mathematical concepts of continuum with the concept of the space-time. (ii) The unification of all forces, field, matter and space-time into one unified force-gravity, that is pure geometry where all the constants of nature emerge form the theory itself. (iii) The Grand unified MU-27 space-time coordinate supersedes the cartesian coordinate system which describes the world of unrelated. (iv) The elucidation of the conceptual structure of space-time, planck constant and planck time and the planck scale often described as foamy through the primordial universal quantum. (v) The rate at which the velocity of the galaxies increase with distance is determined by the system structure of the Grand unified theory MU-27 its principle of continuity and extensity of distance and volume criterion. 1.2 UNIFIED THEORY AND GENERALIZED GROUPS The above notion of quantum space-time is the goal of any unified theory of the physical forces. For instance, from differential geometry, we know that metric can determine the geometry of space(Hawking and Ellis [5]). The idea of generalized groups are tools for constructing unified geometric theory. In this research work, the algebraic structure of generalized groups will be dealt with as follows. Definition 1.1. ([9]) A generalized group G is a non-empty set admitting an operation called multiplication subject to the set of rules given below. (i) (xy)z = x(yz) for all x, y, z ∈ G. A study of some properties of generalized groups 601 (ii) For each x ∈ G there exists a unique e(x) ∈ G such that xe(x) = e(x)x = x (existence and uniqueness of identity element). (iii) For each x ∈ G, there exists x−1 ∈ G such that xx−1 = x−1x = e(x) (existence of inverse element). Example 1.1. Let ( µ ¶ ) 0 0 G = A = , a, b ∈ R and b 6= 0 . a b Then, G is a generalized group and for all A ∈ G, µ ¶ µ ¶ 0 0 −1 0 0 e(A) = a and A = a 1 b 1 b2 b where e(A) and A−1 are the identity and the inverse of matrix A respectively. PROPERTIES OF GENERALIZED GROUPS A Generalized group G exhibits the following properties: (i) for each x ∈ G, there exists a unique x−1 ∈ G. (ii) e(e(x)) = e(x) and e(x−1) = e(x) where x ∈ G. Then, e(x) is a unique identity element of x ∈ G. Definition 1.2. If e(xy) = e(x)e(y), ∀x, y ∈ G, then G is called normal generalized group. 2. LITERATURE REVIEW Mathematicians and physicists have been trying to construct a suitable unified theory, for example twistor theory, isotopies theory and so on. It was known that generalized groups are tools for constructions of a unified geometric theory, electroweak theories are essentially structured on Minkowskian axioms and gravitational theories are constructed on Riemannian axioms. Molaei [9] in his generalized groups established the uniqueness of the identity element of each element in a generalized group and where the identity element is not unique for each element, then a group is formed, some of his results are stated below. Theorem 2.1. For each element x in a generalized group G, there exists a unique x−1 ∈ G. 602 Akinmoyewa, Joseph Toyin Proof. Suppose x ∈ G, then x = xe(x) = x(x−1x) = x(x−1e(x−1))x = (e(x)e(x−1))x and similarly, x = x(e(x−1))e(x) so the uniqueness of e(x) shows that e(x)e(x−1) = e(x−1) = e(x) hence e(x−1) = e(x) (1) Now, let zx = xz = e(x) and yx = xy = e(x). Then z(xy) = z(e(x)), so e(x)y = ze(x) which shows that y = z from (1) The next theorem shows that an abelian generalized group is a group. Theorem 2.2. Let G be a generalized group and xy = yx for all x, y ∈ G. Then G is a group. Proof. We show that e(x) = e(y) ∀ x, y ∈ G. Let x, y ∈ G be given, then (xy)e(y) = xy and e(y)(xy) = e(y)(yx) = yx. Thus, e(xy) = e(y) and by the similar way e(x) = e(xy). Hence, e(x) = e(y). M. Mehrabi, M. R. Molaei and A. Oloomi also considered the idea of generalized subgroups and homomorphisms of generalized groups. Some results on homomorphisms were proved as shown below Theorem 2.3. A non-empty subset H of a generalized group G is a generalized subgroup of G if and only if for all a, b ∈ H, ab−1 ∈ H. Proof. If H is a generalized subgroup and a, b ∈ H, then b−1, ab−1 ∈ H. Conversely, suppose H 6= ∅ and a, b ∈ H, then we have bb−1 = e(b) ∈ H, e(b)b−1 = b−1 ∈ H and ab = a(b−1)−1. Theorem 2.4. Let {Hi|i ∈ I} be a family of generalized group G and Hi 6= ∅. Then Hi is a generalized subgroup of G. −1 Proof. If a, b ∈ H, then a, b ∈ Hi, ∀ i ∈ I, thusab ∈ Hi, ∀ i ∈ I. Hence ab−1 ∈ H and H is a generalized subgroup of G since a, b ∈ H, ab−1 ∈ H. Theorem 2.5. Let G be a generalized group such that a ∈ G. Then, A study of some properties of generalized groups 603 Ga = {x ∈ G : e(x) = e(a)} is a generalized subgroup of G. Furthermore, Ga is a group. Proof. For all a, b ∈ Ga we have (bc)e(a) = bce(c) and e(a)bc = e(b)(bc) = bc, −1 −1 so e(bc) = e(a), we have that e(c) = e(c ). Hence b, c ∈ Ga, and Ga is a generalized subgroup of G. Since the identity function is a constant function on Ga, it is also a group. Definition 2.1. If G and H are two generalized groups and f : g → H, then f is called a homomorphism if f(ab) = f(a)f(b), ∀a, b ∈ G. M. Mehrabi, M.R. Molaei and A. Oloomi [11] stated the following results on homomorphisms of generalized groups. These results are established in this work. Theorem 2.6. Let f : G → H be a homomorphism where G and H are two distinct generalized groups. Then: (i)] f(e(a)) = e(f(a)) is an identity element in H for all a ∈ G. (ii)] f(a−1) = (f(a))−1 (iii) If K is a generalized subgroup of G, then f(K) is a generalized subgroup of H. (iv)] If G is a normal generalized group, then the set {(e(g), f(g)) : g ∈ G} with the product (e(a), f(a))(e(b), f(b)) := (e(ab), f(ab)) is a generalized group denoted by ∪f(G). 2.1. SOME OTHER GENERALIZED CONCEPTS Topological Generalized Groups Molaei [10] introduced topological generalized groups. According to him, a set G is called a topological generalized group if: • G is a generalized group, • G is a Hausdorff topological space and the functions −1 m1 : G → G g 7→ g 604 Akinmoyewa, Joseph Toyin and m2 : G × G → G (gh) 7→ gh are continuous maps.
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