Group Theory | Automorphisms of a Group | Definition with examples | Abstract Algebra | emathsguru

Hello students,
This video lecture on Group theory by Roshan Sir will help you to understand the following topics in Mathematics:
Group Theory | Automorphisms of a Group | Abstract Algebra
1- Definition with examples Automorphisms of a Group
2- Example-1 Show that the mapping
f : I ⟶ I such that f(x) = -x ⍱ x ε I
is an automorphism of the additive group of integers I.
3- Example-2 Show that a⟶a-1 is an automorphism of a group G iff G is abelian.
4- Examples-3 Let G be a group, H a subgroup of G, f an automorphism of G.
Let f(H) = { f(h) : h ε H } . Prove that f(H) is a subgroup of G.
#GroupTheory #Automorphisms_of_a_Group #Abstract Algebra #emathsguru

-----------Content of the Video--------------
00:00 – Intro
01:03 - Definition of automorphism
05:18 - Example 1
12:27 - Example 2
24:25 - example 3
I, Roshan Kumar Jaiswal, welcome you on your YouTube channel eMaths Guru.

This concept is very important to crack TGT/PGT/CSIR NET & equally important in getting good marks in university examinations like B.Sc/M.Sc/B.Tech.
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1- Definition of Groups, Algebraic structure, Grupoid, Monoid

2- abelian groups with solved examples

3-Addition Modulo and Multiplication Modulo

4-Properties of a Group

5- Order of a group and some important theorems

6- Subgroups and some important theorems

7- Cyclic group with examples

8-Cyclic groups theorems

9-Cosets definition and examples

10- Cosets | some important theorems| Lagrange’s theorem

11- Normal subgroups definition with solved examples

12- Normal Subgroups Some Important theorems

13- Quotient Groups with solved examples

14-Homomorphism definition and examples

15- Kernel of homomorphism

16- Isomorphism definition with examples

17- Fundamental theorem of Group homomorphism ( First theorem of Isomorphism )

18- permutation group or symmetric group

19- circular permutation

20- Second theorem of Isomorphism

21- Third theorem of isomorphism

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Thank you
Roshan Kumar Jaiswal
Assistant Teacher
Govt. Basic Education U.P.