As we know magma , semigroup , Monoid , Group and Ring , field from definition let's see some good features and examples of them .
Group example : 1. (Zn,+) Zn = {0,1,...(n-1)} , '+' is addition modulo n
2. (Un,.) Un = {0< x < n | x ⊥ n ( relatively prime )} , '.' is multiplication modulo n
An algebraic structure G = (A,P)| Γ have substructure H = (B,P)| Γ ⇔ B ⊆ A ∧ Γ(B,P) = true .
i.e according to above definition of G,H : (H ≤ G) ⇔ (B ⊆ A ∧ Γ(B,P) = true)
If (G,+) , (H,+) are groups then (H ≤ G) ⇔ ∀ x,y ∈ H x+(-y) ∈ H
left and right cosets of H ≤ G are xH,Hx respectively where x ∈ G
1. Two left/right cosets are either equal or disjoint .
2. If xH = Hx ( H,x are as previous ) then H ◅ G (Normal Subgroup)
Group example : 1. (Zn,+) Zn = {0,1,...(n-1)} , '+' is addition modulo n
2. (Un,.) Un = {0< x < n | x ⊥ n ( relatively prime )} , '.' is multiplication modulo n
An algebraic structure G = (A,P)| Γ have substructure H = (B,P)| Γ ⇔ B ⊆ A ∧ Γ(B,P) = true .
i.e according to above definition of G,H : (H ≤ G) ⇔ (B ⊆ A ∧ Γ(B,P) = true)
If (G,+) , (H,+) are groups then (H ≤ G) ⇔ ∀ x,y ∈ H x+(-y) ∈ H
left and right cosets of H ≤ G are xH,Hx respectively where x ∈ G
1. Two left/right cosets are either equal or disjoint .
2. If xH = Hx ( H,x are as previous ) then H ◅ G (Normal Subgroup)
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