Summary Abstract Algebra - Group, Abelian Group and Semi Group

ABSTARCT
ALGEBRA
 GROUP
 ABELIAN GROUP
 SEMI GROUP

, Group :- Let G be a non- empty set and ‘*’ be a binary
operation on G. Then , the set G with the operation *
denoted by (G,*) is called a Group if and only if the following
conditions are satisfied :-
 If a, b ∈ G then a*b ∈ G
which represents that G shows closure property.
[ Note: * is a binary operation hence it must be
closed]
 If a, b, c ∈ G then a*(b*c) = (a*b)*c
which represents that G show associative property.
 There exists an element e ∈ G such that
a*e = e*a = a ∀ a∈ G
which represents the existence of identity.
 For each element a ∈ G there exist a’ of G such that
a*a’ = a’*a = e
which represents the existence of inverse.
[NOTE : If any one of the conditions does not satisfy then the
set does not form a group]
Abelian Group :- With above four conditions if the set G
satisfies a*b = b*a ∀ a , b ϵ G [ that is commutative]
then (G,*) is said to be Abelian Group or Commutative
Group.