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 }