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January 23rd, 2013, 10:20 AM   #1
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Equivalence Relation

If H be a subgroup of G then prove that the relation R defined on G by aRb iff (a)^(-1).b H for a,bG is an equivalence Relation
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January 24th, 2013, 06:18 AM   #2
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Re: Equivalence Relation

We need R to be,

1. Reflexive : aRa is true because a * a^-1 = e belongs to H since H is a subgroup of G

2. Symmetric : assuming a * b^-1 belongs to H, (a * b^-1)^-1 = b * a^-1 belongs to H. Hence if aRb then bRa.

3. Transitive : If a * b^-1 & b * c^-1 belongs to H then (a * b^-1) * (b * c^-1) = a * c^-1 belongs to H. Hence, if aRb and bRc then aRc.

Hence, R is an equivalence relation.
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February 11th, 2013, 09:20 AM   #3
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Re: Equivalence Relation

Thank u
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