Several Fundamental Theorems in Group Theory

1906 ◽  
Vol 13 (1) ◽  
pp. 10-11
Author(s):  
G. A. Miller
Keyword(s):  
Group Theory ◽  
Fundamental Theorems

scholarly journals The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 411
Author(s):  
Faisal Mehmood ◽  
Fu-Gui Shi ◽  
Khizar Hayat ◽  
Xiao-Peng Yang
Keyword(s):  
Group Theory ◽  
Continuous Functions ◽  
Vital Role ◽  
Ring Theory ◽  
Mathematical Tool ◽  
Algebraic Structures ◽  
Ring Homomorphisms ◽  
Extremum Problems ◽  
Convexity Theory ◽  
Fundamental Theorems

In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we propose fundamental theorems of homomorphisms of M-hazy rings. We also discuss the relation between M-hazy rings and M-hazy ideals. Some important results of M-hazy ring homomorphisms are studied. In recent years, convexity theory has become a helpful mathematical tool for studying extremum problems. Finally, M-fuzzifying convex spaces are induced by M-hazy rings.

From Equations to Structures: Linking Abstract Algebra and High-School Algebra for Secondary School Teachers

2019 ◽  
pp. 517-522
Author(s):  
Josephine Shamash
Keyword(s):  
Field Theory ◽  
High School ◽  
Group Theory ◽  
Secondary School ◽  
Secondary School Teachers ◽  
Abstract Algebra ◽  
School Curriculum ◽  
Algebraic Structures ◽  
High School Curriculum ◽  
Fundamental Theorems

The high-school curriculum in algebra deals mainly with the solution of different types of equations. Modern algebra has a completely different viewpoint and is concerned with algebraic structures and operations. The course Algebra: From Equations to Structures is part of an M.Sc. programme for Israeli secondary school mathematics teachers. It provides an introduction to algebraic structures and modern abstract algebra, and links abstract algebra to the high-school curriculum in algebra. It follows the historical attempts of mathematicians to solve polynomial equations of higher degrees, attempts which resulted in the development of group theory and field theory by Galois and Abel. This approach leads naturally to examining topics and fundamental theorems in both group theory and field theory. Along the historical “journey”, many other major results in algebra in the past 150 years are introduced, and current research in algebra is highlighted. We examine the relevance of the course to the teachers' work.

Group Theory

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