## Group (Defn)

A non-empty set G with binary operation • or is said to be 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 ab =ba

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