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January 23rd, 2013, 09:20 AM  #1 
Newbie Joined: Jan 2013 From: west bengal,india Posts: 22 Thanks: 0  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,b€G is an equivalence Relation

January 24th, 2013, 05:18 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  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. 
February 11th, 2013, 08:20 AM  #3 
Newbie Joined: Jan 2013 From: west bengal,india Posts: 22 Thanks: 0  Re: Equivalence Relation
Thank u


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