GROUP, SUBGROUP, ABELIAN GROUP, NORMAL SUBGROUP AND CONJUGATE NUMBER AND NORMALIZER OF A GROUP IN MODERN ALGEBRA,SOME IMPORTANT TOPICS OF GROUP THEORY IN MODERN ALGEBRA

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

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

3. TABLE OF CONTENTS
1) SOME BASIC TOPICS
2) GROUP
3) ABELIAN GROUP
4) SUBGROUP
5) CONJUGATE NUMBER
6) NORMALIZER OF A GROUP

4. SOME BASIC TOPICS
(a). Closure property:-
Let S be a non-empty set and let * is an
operation in S. If a S and b S then,
Then closure property exist in S w. r. t. operation ‘ ‘ .
Ex:- Set of Natural number, N={ 1,2,3,4 …..}
we take an operation ‘ + ’ .
If a N and b N,
(i). Closure property exit in N w. r. t. addition.
a b S
  , .
a b S
 
*
, ,
a b N a b N
    


 

5. SOME BASIC TOPICS
(ii).Multiplication also satisfied closure property in the
set of N.
(iii). Subtraction does not satisfied closure property in
the set N,
Ex:-
(iv). “ +, -, x ” satisfied closure property in the set of
integer and number.
(b).Binary operation :-Let S be a non-empty set. Let the
operation ‘ * ’ satisfy closure property exist in S.
Then ‘ * ’ is called Binary operation for the set.
Ex:- Addition and Multiplication satisfied the binary
operation for the set of natural number, Integer and
real number etc.
3 ,5 3 5 2
N N N
      

6. SOME BASIC TOPICS
-:Law of Binary operation:-
(C).Associative law:- Let S be a non-empty set and let
‘ * ’ be a binary operation in S.
Let
If .
where
= Associative law exits in S w. r. t binary operation ‘ * ’.
i.e. ‘ * ’ is associative law In S.
, , .
( ) ( )
, ,
a b c S
a b c a b c
a b c S

    


7. SOME BASIC TOPICS
(d). Commutative law:- Let S be a non-empty set and
let * is a binary operation in S.
let
If
then commutative law exist in S w. r. t binary
operation *
i.e. ‘ * ’ is commutative in S.
, .
, .
a b S
a b b a a b S

    

8. SOME BASIC TOPICS

9. SOME BASIC TOPICS

10. GROUP

11. GROUP
:- Let G be a non-empty set and and * be a
binary operation defined
on it, then the structure
(G,*) is said to be a
group, if the following
axioms are satisfied,
(i) Closure property :- a *
b G, a,b G
(ii) Associativity :- The operation * is
associative on G. i.e.
a * (b * c) = (a* b) * c, a,b,c G
 



12. GROUP

13. ABELIAN GROUP
•

14. ABELIAN GROUP
•

15. ABELIAN GROUP
•

16. •SUB-GROUP
Definition:- A non-empty subset H of a group (G,
*) is said to be subgroup of G, if
(H, *) is itself a group.
Example:- [{1,-1}, .] is a
subgroup of [{1.-1}, i,-i}.]
Criteria for a subset to be a
subgroup:-A non-empty subset
H of a group G is a subgroup of
G if and only if
(i ) a,b H => ab H
-1 -1






17. C0NJUGATE NUMBER
Conjugate number and relation of conjugate
:-let G is a group. Let a and b are two elements
of G. if there exist some ‘x’ in G.
such that:
‘a = x-1 b.x’ then a is called
conjugate to b. In symbol it is written as a~ b
and this relation in G is called relation of
conjugacy.

18. NORMALIZER OF A GROUP
• Let G is a group. Let a G N(a) is the set of all


19. Professor Einstein Writes in Appreciation of a
Fellow –Mathematics:-
ure mathematics is, in its way, the poetry of logical
ideas. One seeks the most general ideas of operation
which will bring together in simple, logical and unified
form the largest possible circle of formal relationships.
In this effort toward logical beauty spiritual formulas
are discovered necessary for the deeper penetration
into the laws of nature.
ALBERT EINSTIN
(Princeton University, May 1, 1935)

20. THANK YOU