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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 lawa,b ϵ G ⇨ a ∗ b ϵ G2.Associative lawa,b,c ϵ G ⇨(a∗b)∗c =a∗(b∗c)3.The existence of Identitythe exists e ϵ G such that a ∗ e = e ∗a = a, ⩝ a ϵ G4.The existence of Inversethere exists aʹϵ G such that aʹ∗a = a∗aʹ, ⩝ a ϵ G.Commutative Group or Abelian GroupLet 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∗aQuasi GroupLet G be a non empty set then G satisfies closure law onlya,b ϵ G ⇨ a ∗ b ϵ GSemi-GroupLet 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)MonoidLet G be a non empty set then G satisfies closure law , associative law and identity existence onlyclosure: a,