Q1- Prove that ∼H is an equivalence relation on G. That is, ∼H satisfies: (i) For every g ∈ G, g ∼H (Reflexivity) (ii) For all g1, g2 ∈ G, If g1 ∼H g2, then g2 ∼H g1. (Symmetry) (iii) For all g1,g2,g3 ∈G, If g1 ∼H g2 and g2 ∼H g3,then g1 ∼H g3. 
(Transitivity) Cosets. For g ∈ G, let gH = {g · h : h ∈ H}.
The set gH is known as the left coset of H containing g.   Q2- Let φ : G → H be a group homomorphism. (a) Prove that Ker(φ) is a normal subgroup of G. (a) Prove that Im(φ) is a subgroup of G. Is it normal? When?

Question

Q1- Prove that ∼H is an equivalence relation on G.

That is, ∼H satisfies:

  • (i) For every g ∈ G, g ∼H (Reflexivity)
  • (ii) For all g1, g2 ∈ G, If g1 ∼H g2, then g2 ∼H g1. (Symmetry)
  • (iii) For all g1,g2,g3 ∈G, If g1 ∼H g2 and g2 ∼H g3,then g1 ∼H g3. 
(Transitivity)

Cosets. For g ∈ G, let gH = {g · h : h ∈ H}.
The set gH is known as the left coset of H containing g.

 

Q2- Let φ : G → H be a group homomorphism.

(a) Prove that Ker(φ) is a normal subgroup of G.

(a) Prove that Im(φ) is a subgroup of G. Is it normal? When?

 

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