This document discusses key concepts in abstract algebra including groups, subgroups, normal subgroups, abelian groups, rings, and fields. It provides definitions and examples for each concept. Groups are defined as sets with a binary operation that satisfy closure, associativity, identity, and inverse properties. Subgroups are subsets of a group that are also groups. Normal subgroups are subgroups where applying the group operation to a subgroup element and any group element results in another subgroup element. Abelian groups are groups where the group operation is commutative. Rings are algebraic structures with two binary operations that satisfy properties including being abelian groups under addition and semi-groups under multiplication while satisfying distributivity. Fields are non-trivial rings where multiplication is also commutative.

1. NATIONAL MATHEMATICS DAY
THE MAN WHO KNEW INFINITY
SRINIVASA RAMANUJAN LYENGAR
(Best know as S. RAMANUJAN)
(22nd DECEMBER 1887 – 26th APRIL 1920)

2. ABSTRACT ALGEBRA
BY
SONU KUMAR
M.Sc.-4th Semester
Department of Mathematics
R.N. College, Hajipur (Vaishali)
(B.R.A. Bihar University, Muzaffarpur)

3. OVERVIEW
 GROUP
 SUB-GROUP
 NORMAL SUB-GROUP
 ABELlAN GROUP
 RING
 FIELD

4. GROUP
Group:-Let G be a non-empty set and (G,*) be
binary operation then it has satisfied the following
properties…
(i).Closure law :- if a ,b G than a*b G is called
closure law. And
(ii).Associative law:- if a,b,c G than
(a * b) * c = a * (b * c),
 

, , .
 
a b c G
, .
 
a b G

5. GROUP
(iii).Existence of Identity:- If a G and e G
then
a * e = a = e * a ,
(iv).Existence of Inverse :- If a G and a-1 G
then
a * a-1 = e = a-1 * a,
 
, .
 
a e G
1
, , .

 
a a e G

 

6. SUB-GROUP
Sub-group:-Let (G,*) and (H,*) are two non-
empty group and if H is a subset of G than (H,
*) IS called subgroup of (G, *).
I = { 0, ± 1, ±2, ±3 ……. }
E = { 0, ±2, ±4, ±6 …... }
(I,+) is group but E is a proper subset of I,
So E is a sub group of I.



7. NORMAL SUBGROUP
Normal subgroup:- Let (G, *) is a
group and (N,*) is a subgroup of G
then N is said to be normal
subgroup of G.
g.n.g-1 N, for all g G, n N.

 

8. ABELIAN GROUP
Abelian group:- An abelian group also called a
commutative group is a group in which the
result of applying the group operation to two
group elements does not depend on the order
in which they are written, that is the group
operation in commutative.

9. RING
Ring:-An order triple (R,+,x) is called a ring,
if it satisfy the following axioms:
(i) R is abelian group under addition.
(ii) R is semi - group under multiplication.
(iii) Distributive law holds in R.

10. FIELD
Field:- A field is a non-trivial division
ring whose ring product is
commutative. Thus, let (F,+,×) be an
algebraic structure.

11. THANK YOU