normal351 - MAT 312/AMS 351 Notes and exercises on normal...

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Unformatted text preview: MAT 312/AMS 351 Notes and exercises on normal subgroups and quotient groups. If H is a subgroup of G , the equivalence relation ∼ H is defined be- tween elements of G as follows: g 1 ∼ H g 2 ⇔ ∃ h ∈ H, g 1 = g 2 h. Proposition 1. This is indeed an equivalence relation. Proof: The three properties: reflexive, symmetric, transitive corre- spond to the three properties of a subgroup: H contains the identity element e of G , H contains inverses of all its elements, H is closed under composition. • For any g ∈ G , since e ∈ H and ge = g , it follows that g ∼ H ge = g , so the relation ∼ H is reflexive. • If g 1 ∼ H g 2 , ∃ h ∈ H, g 1 = g 2 h . Since h − 1 must then also belong to H , and g 1 h − 1 = g 2 hh − 1 = g 2 , it follows that g 2 ∼ H g 1 , so the relation ∼ H is symmetric. • If g 1 ∼ H g 2 and g 2 ∼ H g 3 , then ∃ h ∈ H, g 1 = g 2 h , and also ∃ h ′ ∈ H, g 2 = g 3 h ′ . Since then hh ′ ∈ H , and g 1 = g 2 h = ( g 3 h ′ ) h = g 3 ( h ′ h ), it follows that g 1 ∼ H g 3 ; so the relation ∼ H is transitive. For the ∼ H equivalence class of the element g ∈ G we have the suggestive notation gH (since every element of that equivalence class is gh for some h ∈ H ); This equivalence class is called the left H-coset of g ; “left” because gH is obtained by multiplying every element of H on the left by g . Note that the left H-coset of the identity e is H itself....
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normal351 - MAT 312/AMS 351 Notes and exercises on normal...

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