Group Theory

Group (Defn)  

A non-empty set G with binary operation • or ∗ is said to be a group if G satisfies groups laws,  〈group denoted by (G,∗)〉

Groups laws  

      1.Closure law  

         a,b ϵ G ⇨ a ∗ b ϵ G

2.Associative law

     a,b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c)

3.The existence of Identity

     the exists e ϵ G such that a ∗ e = e ∗a = a, ⩝ a ϵ G

4.The existence of Inverse

     there exists aʹϵ G such that aʹ∗a = a∗aʹ, ⩝ a ϵ G.

Commutative Group or Abelian Group

Let G be a group with binary operation ∗  〈i.e (G,∗) be a group〉

then for every a,b ϵ G such that a∗b =b∗a

Quasi Group

      Let G be a non-empty set then G satisfies closure law only

     a,b ϵ G ⇨ a ∗ b ϵ G

     Semi-Group

     Let G be a non-empty set then G satisfies closure law and

     associative law only

     closure: a,b ϵ G ⇨ a ∗ b ϵ G

     associative: a,b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c)

     Monoid

      Let G be a non-empty set then G satisfies closure law ,

      associative law and identity existence only

      closure    : a,b ϵ G ⇨ a ∗ b ϵ G

      associative : a,b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c)

      identity   :  the exists e ϵ G such that a ∗ e = e ∗a = a, ⩝ a ϵ G

 

    Hints:

Groups/Laws	Closure	Associative	Identity	Inverse	Commutative
Quasi	Yes	No	No	No	No
Semi-group	Yes	Yes	No	No	No
Monoid	Yes	Yes	Yes	No	No
Group	Yes	Yes	Yes	Yes	No
Abelian group	Yes	Yes	Yes	Yes	Yes