Knowledge in Group theory
Subgroup theorems
Theorems are If H and K are subgroups of a group G then O(HK) =O(H).O(K) and also |HK| =|H||K| /|HUK| and also O(H) > equal to √O(G) and vice versa
Ishant Goel
Cosets
Definition of cosets with example and definition of index of subgroups and the most important theorem discussed here i.e Lagrange's theorem
Normalizer
Definition of double cosets amd difference between cosets and double cosets and a theorem that double cosets defines a partition in a group and the definition of normalizer and that normalizer is a subgroup
Centralizer
Definition of centralizer of a group And center of a group... Prove that centralizer of a group os subgroup then the definition of conjugate or transform in a group and the relation of conjugacy is an equivalence relation and some questions
Self-conjugate and conjugacy classes
Definition of self-conjugate and conjugacy classes and the result that the number of elements in a conjugacy classes is equal to the index of normalizer
Class Equation
These notes includes definition of class Equation and p-group and the theorem that the center of p-group is non-trivial And every group of order p^n is non-trivial center
Conjugate subgroup
Definition of conjugate subgroup and a theorem that if H is a subgroup of G and K is conjugate to H then K is subgroup of G and another result that conjugate subgroups are isomorphic and the theorem of left coset
Normal Subgroup
Definition of normal Subgroup of a group G, the most important thereom that kernel is a normal Subgroup.... In an abelian group, each subgroup is normal... normal Subgroup means left coset = right coset
Quotient group
The another name of quotient group is factor group, it's definition, properties and a most important result that If Phi is defined in a mapping in a group then Phi is epimorphism
First Isomorphism thereom
This is the most important thereom of group Homomorphisms that is first Isomorphism theorem, statement of the theorem is also most important and can be asked in exam... Also it's proof and a Corollary us discussed here
second and third isomorphism theorem
These are the second and third isomorphism theorem .... Actually isomorphism is one-to-one onto correspondence of functions and the statement and proof of the theorems is based on this fact
Automorphism
Definition of endomorphism and automorphism... automorphism is Homomorphism as well as bijective... The proof of set kf automorphism is also a group and at last, there is a lemma conjugation as an automorphism