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Showing posts with the label Group Theory

Group Theory

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        t he exists e ϵ G such that a ∗ e = e ∗a = a, ⩝ a ϵ G 4.The existence of Inverse        there exists a ʹϵ G such that a ʹ∗a = a∗ a ʹ, ⩝ 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      c losure: a, b ϵ G ⇨ a ∗ b ϵ G       a ssociative: a, b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c)      Monoid        Let G be a non empty set then G satis