Basics of Group theory

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

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+bєG

7. Examples of Subgroups

8. 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>

9. 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.

10. • 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.