CLOSURE PROPERTY, ASSOCIATIVE LAW,BINARY OPERATION, COMMUTATIVE LAW,DISTRIBUTIVE LAW IN GROUP THEORY, GROUP, NORMAL SUBGROUP, CONJUGATE NUMBER, SUBGROUP GROUP, ABELIAN GROUP AND THEIR APPLICATION

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

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

3. TABLE OF CONTENTS
 SOME BASIC TOPICS
(a) Closure property (b) Binary operation (c) Associative law
(d) Commutative law (e) Distributive law (f) Identity element
(g) Inverse element .
 GROUP
 ABELIAN GROUP
 SUBGROUP
 NORMAL SUBGROUP
 CONJUGATE NUMBER
 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 real 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
(e). Distributive law :- Let S be a non- empty set. Let
* and ° are two binary operation in S.
(f). Identity element :- Let S be a non-empty set and
let * be a binary operation in S.
     
Let a,b,c ∈
S
If a * boc = a *b o a * c ∀a,b, c ∈
S.
here * is distributive over ' o'.
Let a ∈
S andlet e ∈
S
If a *e =e *a =a ∀a ∈
S.
then e is called identity element inS w. r. t ' * '.

9. SOME BASIC TOPICS
(g). Inverse element :- Let S be a non-empty set
and let * be a binary operation in S.
-1
-1 -1
Let e ∈
S, leta ∈
S andb ∈
S
suchthat, if a *b =b * a = e
then bis calledinverse ofa w.r.t ' * '.
inverse of a is denoted by a
a * a = a * a =e ∀a, e ∈
S.

10. GROUP
 GROUP :- A non-empty set G together with an
operation ‘ * ‘ is called group if,
(i)Closure property:-
(ii)Associative law :-
Let a,b ∈
G.
If a *b ∈
G ∀a,b ∈
G.
⇒Closure property exist inG w.r. t ' * '.
   
Let a,b,c ∈
G
If a *b *c =a * b*c ∀ a,b, c ∈
G.
⇒associative law exist in G w.r.t ' * '.

11. GROUP
(iii).Identity element :-
(iv). Inverse element :- for each element a G.
those exit an element a-1
such that,
Let a ∈
G and lete ∈
G.
If a *e =e *a =a ∀a ∈
G.
then e is calledis identity elementin G w.r.t ' *'.
-1 -1
-1
a * a = a * a =e.
where a is the inverse
element exist in G w.r.t ' * '.

12. ABELIAN GROUP
Definition :- Let a non-empty set G together with an
Operation * satisfy the following condition is called
an abelian group if,
   
Let a,b G
a b G a,b G.
satisfy closure property in G.
b c a b c a,b,c G.
Associative law satisfly in G w.r .
.t ' '

    
 
      
 
( i) Closure property : -
( ii) Associative law: - Let a,b,c ∈
G.
If a

13. ABELIAN GROUP
1
1 1
1
(iii)
(iv)
.
(v)
Let a G and e G
If a e e a a a,e G.
e is called identity element in G.
Let a G and e G
a G then a G
If a a a a e
a is called inverse of a in G each
elements exist itsinverse in S

 

 
     

 
 
   

Identity element: -
Inverse element: -
Let a,b G
If a b b a a,b G.
then satisfy commutative law in G.

    

Cummutative law: -

14. SUB-GROUP
Definition:- Let G be a group and H is the subset of G.
If H is also a group w. r. t. the same operation as that
of G then H is called sub-group of G.
Ex:- we know that set of integer that is I is a group
w. r. t. addition , i.e. (I, +) is a
group.
we also know that set of even
integer E is also a group
w. r. t. addition, i.e. (E,+) is a
group. Where I and E are group
w. r. t. addition and
E I.


15. NORMAL SUBGROUP
NORMAL SUBGROUP:- let G be a group. Let H is
the subgroup of G. Then H is called normal
subgroup of G,
PROVE THAT EVERY SUBGROUP OF AN ABELIAN
GROUP IS NORMAL SUBGROUP.
Soln :- let G is an abelian group and let H is the
subgroup of G, then i have to prove that H is the
normal subgroup of G.
i.e. I have to show that,
-1
If x.h.x ∈
H ∀
x ∈
G and h ∈
H.

16. NORMAL SUBGROUP
 
   
 
 
 
 
-1
-1 -1
-1
-1
-1
-1
i.e. i have to show that,
x.h.x ∈
H ∀x ∈
G and h ∈
H.
From associstivelawinG.
Now, x.h.x = x.h .x
where x,h,x ∈
G
= h.x .x ∵
G is an abelian.
=h. x.x
=h.e
=h∈
H
⇒ x.h.x ∈
H
Hence His the normal subgroup of G.

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 those element of G
which commute 'a'. thenN( a)is
called the normalizer of the
element 'a' andN( a)∈
G
whichalso canbe written as,
N( a)=
{ x
Definitio
∈
G : x.a =
n: -
a.x }

19. THANK YOU