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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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