Basics of Group Theory in Mathematics

 
Basics of Group Theory
B.Sc-II Sem –III
Unit-I
 
By
Mr.M.S.Wavare
Department of Mathematics
Rajarshi Shahu Mahavidyalaya, Latur
(Autonomous)
 
Introduction
 
Set: The collection of well defined objects is
called set.
    Examples:-
     A={1,2,3,4,5}
     N={1,2,3,………}
     W={0,1,2,3,….}
      I={….,-3,-2,-1,0,1,2,3,….}
     R={real numbers}
 
 Semigroup & Monoid
 
The non empty set G  and * be a binary
operation on G is said to be semigroup if it
satisfied associative property.
    Example: N={set of natural numbers} is a semi
group.
A semigroup (G,*) having an identity element
is called a monoid.
    Example: I
+
 ={0,1,2,3……}
 
Definition of Group
 
 
 
 
 
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C
l
o
s
u
r
e
:
 
f
o
r
 
a
l
l
 
a
,
b
 
i
n
 
G
,
 
a
+
b
є
G
 
 
Examples of Subgroups
 
Cyclic Group
 
A group G is cyclic if there is an element a in G,
such that every element of G is some power of
a
The group G is said to be generated by a, and
a is called as generated of G.
Examples: Z is an infinite cyclic group and
Z=<1,-1>
 
Normal Subgroup
 
A subgroup N of a group G is said to be a
normal subgroup of gG if gN=Ng for all g in G.
 
Every subgroup of abelian group is normal.
 
Every subgroup of index 2 is normal.
 
Simple Group: A simple group is a group of
order greater than 1 whose only normal
subgroups are the identity subgroup and
group itself.
 
Factor Group: Let H be a normal subgroup of a
group G. Then, G/N be the set of all coset of H
in G is a group with respect to the binary
operation defined by aHbH=abH, for all aH,bH
in G/H. It is called as Factor Group.
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Introduction to sets in mathematics, concepts of semigroup and monoid, definition of a group, examples of subgroups, cyclic groups, normal subgroups, simple groups, and factor groups. Exploring fundamental principles in group theory.


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  1. Basics of Group Theory B.Sc-II Sem III Unit-I By By Mr.M.S.Wavare Mr.M.S.Wavare Department of Mathematics Department of Mathematics Rajarshi Rajarshi Shahu Shahu Mahavidyalaya Mahavidyalaya, , Latur (Autonomous) (Autonomous) Latur

  2. Introduction Set: The collection of well defined objects is called set. Examples:- A={1,2,3,4,5} N={1,2,3, } W={0,1,2,3, .} I={ .,-3,-2,-1,0,1,2,3, .} R={real numbers}

  3. Semigroup & Monoid The non empty set G and * be a binary operation on G is said to be semigroup if it satisfied associative property. Example: N={set of natural numbers} is a semi group. A semigroup (G,*) having an identity element is called a monoid. Example: I+={0,1,2,3 }

  4. Definition of Group iv) Closure: for all a,b in G, a+b G

  5. Examples of Subgroups

  6. Cyclic Group A group G is cyclic if there is an element a in G, such that every element of G is some power of a The group G is said to be generated by a, and a is called as generated of G. Examples: Z is an infinite cyclic group and Z=<1,-1>

  7. Normal Subgroup A subgroup N of a group G is said to be a normal subgroup of gG if gN=Ng for all g in G. Every subgroup of abelian group is normal. Every subgroup of index 2 is normal.

  8. Simple Group: A simple group is a group of order greater than 1 whose only normal subgroups are the identity subgroup and group itself. Factor Group: Let H be a normal subgroup of a group G. Then, G/N be the set of all coset of H in G is a group with respect to the binary operation defined by aHbH=abH, for all aH,bH in G/H. It is called as Factor Group.

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