Group Theory: A Comprehensive Overview
Explore the concepts of groups, generators, group presentations, homomorphisms, isomorphisms, and Baumslag-Solitar groups in this detailed explanation. Dive into the fundamental principles and properties that define group theory, offering insights into mathematical structures and group comparisons.
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Presentation Transcript
oups What is a Group A Group G is a set taken with a binary operation * that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied 1. Closure Property 2. Associativity 3. Existence of Identity Element 4. For every element there should Exists a of Inverse inside the group itself. Group Generator
Group Generator A generator is (are) element(s) of a group which can be used to generate whole Group itself when combined via the group operation For example consider the group of integers modulo 4, G={0,1,2,3} , * (addition modulo 4 as group s binary operation) now notice that 1 is the generator element why? 1^0 = 1 1*1 = 1+1 mod 4 = 2 1*1*1 = 1+1+1 mod 4 = 3 1*1*1*1 = 1+1+1+1 mod 4 = 0 Generator Presenta
or Presentationofa group How to Represent a Group Presentation is a method to represent/specify a group. Presentation of group G consist of 1. A set of Generators (S) . It is subset of Group Set. 2. A set of Relation or simply some Relation governing the condition of combination of the elements (called as relator) For example one can see that in previous example of ?4 , (addition modulo 4) can be generated by using the element 1 thus its representation is 1 | 1? where 1^n means 1*1*1..*1 n times Similarly one can see the ?+ ?+ can be represented as ?,? ?? = ?? here group operation is multiplication and relation is the commutativity of multiplication H
up What is Homomorphism Homomorphism Homomorphism is a way to compare the structural similarity of two groups, it preserves the operation. For example consider the group 1. G, * 2. H, # Now let x,y ? ? ???? ? = ? ? ? ? now consider a function ?:? ? which maps every element of G to H and may or may not be injective or surjective. Now let 1. ? ? ? ? ? 2. ? ? ? ? ? 3. ? ? ? ? ? Then what we want is that the structure is preserved that is ? ? = ? ? ? #? ? = ? ? = ?(? ?) Isomor Moreover one can see that any homomorphism maps identity to identity and Inverses to inverses .
m Isomorphism Isomorphism One may define isomorphism as a bijective homeomorphisms Or equivalently one may define that a morphism whose inverse is also a morphism Epimorphism One may define Epimorphism as a surjective homeomorphism. Hopfian Group FreeGro A group is called Hopfian if every epimorphism from the group to itself is an isomorphism.
Baumslag-Solitar Groups Baumslag Solitar groups The Baumslag Solitar groups are a particular class of two- generator one-relator groups and are used as a counter example or example of different types/classes of group. They mark boundary between different types/classes. They are defined very simply as for m, n being positive integers: ?? ?,? = ?,? ???? 1= ?? Moreover when ? = 1, ? = 1 this group ?? 1, 1 gives fundamental group of Klein Bottle For example: Baumslag-Solitar Groups BS(3,2) is Non-Hopfian
Baumslag-Solitar Groups Baumslag Solitar groups I found the following theorem without proof while reading some text online. Theorem: The groups BS(m, n) are Hopfian if m and n have same prime factor The basic importance of Baumslag-Solitar Groups which I understand is its versatility to be used as example and counter- example for different types of groups.
