A computer proof of relations in a certain class of groups

1991 ◽  
Vol 117 (1-2) ◽  
pp. 109-114
Author(s):  
Edmund F. Robertson ◽  
Kevin Rutherford
Keyword(s):  
Group Theory ◽  
Special Cases ◽  
Computer Proof

SynopsisA gp-toolkit consisting of computer implementations of various group theory methods, in particular a Tietze transformation program, was designed. Special cases of a conjecture were solved by the gp-toolkit. Examination of the method used by the gp-toolkit to deduce relations showed that a general approach had been employed. We present a proof verifying that the conjecture is true which is a straightforward generalisation of the method discovered by the gp-toolkit.

scholarly journals Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 321 ◽  
Author(s):  
Mehmet Çelik ◽  
Moges Shalla ◽  
Necati Olgun
Keyword(s):  
Group Theory ◽  
Significant Role ◽  
Classical Group ◽  
Algebraic Structures ◽  
Algebraic Systems ◽  
Homomorphism Theorem ◽  
Special Cases ◽  
Isomorphism Theorems

In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related.

scholarly journals Norton Algebras of the Hamming Graphs via Linear Characters

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Group Theory ◽  
Automorphism Group ◽  
Nonassociative Algebra ◽  
Bilinear Forms ◽  
Distance Regular Graph ◽  
Finite Abelian Groups ◽  
Hamming Graph ◽  
Large Subgroup ◽  
Special Cases ◽  
Hamming Graphs

The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.

Some properties of repetends

2003 ◽  
Vol 87 (510) ◽  
pp. 437-443
Author(s):  
N. J. Armstrong ◽  
R. J. Armstrong
Keyword(s):  
Group Theory ◽  
General Condition ◽  
Number System ◽  
Special Cases ◽  
The Subject

We wish to discuss some aspects of repetends, the repeating sequence of digits in the expansion of a fraction (for illuminating introductions to the subject see [1, 2]). For the most part we restrict consideration here to fractions with a prime denominator. But we do consider the general condition for the length of repetends and examine some special cases when the base of the number system is varied. An illustration of the use of other bases than 10 is given. Then we consider the multiplication of repetends and show a connection with group theory, giving an old result by a new twist.

Electron Beam Damage of Biomolecules Assessed by Energy Loss Spectroscopy

1978 ◽  
Vol 36 (3) ◽  
pp. 61-69
Author(s):  
M. Isaacson ◽  
M.L. Collins ◽  
M. Listvan
Keyword(s):  
Electron Microscopy ◽  
Radiation Damage ◽  
Structural Integrity ◽  
Analytical Electron Microscopy ◽  
High Dose ◽  
Dose Rates ◽  
Special Cases ◽  
Analytical Electron ◽  
Atom Migration ◽  
Beam Damage

Over the past five years it has become evident that radiation damage provides the fundamental limit to the study of blomolecular structure by electron microscopy. In some special cases structural determinations at very low doses can be achieved through superposition techniques to study periodic (Unwin & Henderson, 1975) and nonperiodic (Saxton & Frank, 1977) specimens. In addition, protection methods such as glucose embedding (Unwin & Henderson, 1975) and maintenance of specimen hydration at low temperatures (Taylor & Glaeser, 1976) have also shown promise. Despite these successes, the basic nature of radiation damage in the electron microscope is far from clear. In general we cannot predict exactly how different structures will behave during electron Irradiation at high dose rates. Moreover, with the rapid rise of analytical electron microscopy over the last few years, nvicroscopists are becoming concerned with questions of compositional as well as structural integrity. It is important to measure changes in elemental composition arising from atom migration in or loss from the specimen as a result of electron bombardment.

Decoration technique and surface physics

1978 ◽  
Vol 36 (1) ◽  
pp. 436-437 ◽  
Author(s):  
H. Bethge
Keyword(s):  
Diffusion Path ◽  
Special Importance ◽  
Surface Physics ◽  
Step Structure ◽  
Initial Stage ◽  
Special Cases ◽  
Atomic Surface ◽  
The Mean ◽  
Bulk Structure ◽  
Decoration Technique

Besides the atomic surface structure, diverging in special cases with respect to the bulk structure, the real structure of a surface Is determined by the step structure. Using the decoration technique /1/ it is possible to image step structures having step heights down to a single lattice plane distance electron-microscopically. For a number of problems the knowledge of the monatomic step structures is important, because numerous problems of surface physics are directly connected with processes taking place at these steps, e.g. crystal growth or evaporation, sorption and nucleatlon as initial stage of overgrowth of thin films.To demonstrate the decoration technique by means of evaporation of heavy metals Fig. 1 from our former investigations shows the monatomic step structure of an evaporated NaCI crystal. of special Importance Is the detection of the movement of steps during the growth or evaporation of a crystal. From the velocity of a step fundamental quantities for the molecular processes can be determined, e.g. the mean free diffusion path of molecules.

Group Theory

2009 ◽  
Author(s):  
Pierre Ramond
Keyword(s):  
Group Theory
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